LIBRARY 

OF   THE 

UNIVERSITY  OF  CALIFORNIA. 
•^Received         (AN      6    1893   •  l89 

Accessions  No. ^-f^QO^t.  Class  No. 
\ 


GRUBE'S 
METHOD  OF  TEACHING 

ARITHMETIC 


Explained  and  Illustrated.    Also  the  im- 
provements upon  the  method  made 
by  the  followers  of  Grube 
in  Germany. 


LEVI  SEELEY,  A.M.,  PH.D. 


*'  THAT  MAN  WILL  BE  A  BENEFACTOR  OF  HIS  RACE  WHO  SHALL  TEACH  us 

HOW  TO   MANAGE   RIGHTLY  THE   FIRST  YEARS   OF  A  CHILD'S  EDUCATION.'* 

Garfiefd. 


NEW  YORK  AND  CHICAGO: 

E.   L  KELLOGG  &  CO. 


[US  17  BE  SIT  7) 


L_- 


COPYRIGHT  BY 

E.  L.  KELLOGG  &  CO., 

1888. 


PREFACE. 


THERE  is  a  widespread  feeling  among  American 
teachers  that  there  is  need  of  better  methods  of  teaching 
Number,  especially  in  the  primary  classes.  During  the 
last  few  years,  the  Grube  System,  having  been  introduced 
into  a  few  schools  and  discussed  at  teachers'  institutes  and 
in  educational  journals,  has  attracted  the  attention  of 
thoughtful  educators  in  various  parts  of  our  land. 
Many  of  the  later  Arithmetics  have  devoted  a  few  pages 
in  outlining  this  system  or  in  giving  a  few  hints  in  regard 
to  it.  The  excellent  results  apparent  in  those  schools 
that  have  tried  the  system,  the  enthusiasm  of  its  ad- 
herents, and  the  belief  that  it  is  based  on  sound  philo- 
sophical principles,  have  created  a  desire  to  a  better 
understanding  of  it.  The  design  of  this  little  book  is  to 
give  a  plain,  concise  exposition  of  the  Grube  theory,  and, 
at  the  same  time,  to  illustrate  the  method  of  teaching 
Number  in  accordance  with  it.  It  is  intended  to  be  a 
helpful  book  for  the  primary  teacher. 

This  book  is  not  simply  a  translation  of  Grube's  treatise, 
nor  is  it  the  Grube  system  exclusively;  it  includes  all  of 
that  system,  and  in  addition  the  latest  and  best  ideas  of  the 
disciples  of  Grube  in  Germany,  whose  works  were  studied 


4  PREFACE. 

and  whose  personal  acquaintance  was  made  by  the 
author  in  their  school-rooms  and  in  their  educational  as- 
sociations during  a  three  years'  study  of  the  German 
schools. 

In  the  preparation  of  this  book  I  have  examined 
with  care  the  following  works  :  Grube's  ' '  Leitfaden  fur 
das  Rechnen  in  der  Elementarschule"  (the  original  ex- 
position of  Grube's  system),  German  works  on  Arith- 
metic by  Brautigam,  Gopfert,  Lincke,  Schellen,  Bert- 
helt  und  Petermann,  Rein's  "Theorie  und  Praxis,"  Sol- 
dan's  " Grube's  Method,"  Indianapolis  School  Manual 
of  1876,  Quincy  (Mass.)  Course  of  Study  of  1879,  anc* 
other  treatises. 

LEVI  SEELEY. 

LAKE  FOREST,  ILL.,  1888. 


SKETCH  OF  GRUBE. 


AUGUST  WILHELM  GRUBE  was  born  in  Wernigerode  at 
the  foot  of  the  Harz  Mountains,  Germany,  on  the  i6th  of 
December,  1816.  His  father  was  a  tailor,  and  August  was 
his  only  child.  He  commenced  school  when  four  years 
of  age,  and  very  early  decided  to  devote  himself  to  teach- 
ing. Grube  often  said  in  later  years  that  it  was  his  love 
for  his  teacher  that  awakened  in  him  the  wish  to  become 
a  teacher. 

When  eight  years  of  age  he  entered  the  Lyceum  of  his 
native  city,  where  he  remained  till  his  fifteenth  year, 
after  which  he  entered  the  Teachers'  Seminary  at  Weis- 
senfels  near  Leipzig.  In  1836,  when  twenty  years  of  age, 
he  completed  his  work  here,  and  obtained  a  testimonial 
which  stated  that  Grube  was  well  fitted  to  teach  in  the 
best  grade  of  schools.  After  teaching  a  short  time  in  a 
public  school,  he  took  the  position  of  family  teacher 
(Hauslehrer)  in  the  family  of  a  count  in  the  province  of 
Posen.  A  like  position  in  the  family  of  a  wealthy  manu- 
facturer near  Bregenz  occupied  his  time  until  he  gave  up 
teaching  and  devoted  himself  to  authorship.  He  died 
January  27,  1884. 

"  Grube  was  one  of  the  most  fruitful  and,  at  the  same 
time,  most  important  pedagogical  authors  of  the  present 
time;  a  man  endowed  with  philosophical  penetration 
and  sound  knowledge,  great  from  inclination  and  char- 


6  SKETCH  OF  GRUBE. 

acter,  likewise  rich  in  the  experiences  of  life  and  of  the 
schoolroom.  He  has  by  means  of  his  writings  exercised 
an  extensive,  blessed  influence  upon  the  educators  of  our 
time." 

His  works  cover  all  departments  of  pedagogics.  From 
the  many  we  name  "  Pedagogical  Studies  and  Criticisms" 
(Padagogische  Studien  und  Kritiken),  in  one  part  of  which 
he  discusses  "  Darwinism  and  its  Consequences,"  taking  a 
stand  against  Darwin.  Especially  has  Grube  rendered 
great  service  to  the  young  by  his  "  Geographical  Char- 
acter-pictures" (Geographischen  Charakterbilder),  "  Bi- 
ographies from  Natural  Science"  (Biographien  aus  der 
Naturkunde),  "Character-pictures  from  History  and  Tra- 
dition" (Charakterbilder  aus  der  Geschichte  und  Sage). 
Grube  was  the  forerunner  of  new  methods  of  teaching 
geography  in  Germany.  He  opposed  the  practice  of 
making  the  study  of  geography  a  memorizing  of  num- 
bers and  facts,  and  arranged  the  material  to  be  taught  so 
that  it  could  be  used  to  advantage.  He  connected  the 
teaching  of  geography  with  pictures  of  the  landscape,  with 
productions,  temperature  of  the  country,  and  would  show 
how  the  customs,  religion,  government,  history,  and  hap- 
piness of  the  people  are  related  to  and  dependent  upon 
the  country. 

But  of  particular  interest  to  us  in  connection  with  this 
work  is  Grube's  "  Guide  for  Reckoning  in  the  Elementary 
School,  according  to  the  Principles  of  an  Inventive 
Method"  (Leitfaden-fiir  das  Rechnen  in  der  Elementar- 
schule,  nach  den  Grundsatzen  einer  heuristischen 
Methode).  This  book  marked  an  epoch  in  the  teaching 
of  Number  in  Germany  and  has  exerted  a  wide  influence 
on  American  teaching. 


CONTENTS. 


PAGE 

I.  SKETCH  OF  GRUBE  ......       9 

II.  INTRODUCTION          .  .  .  .  .  .11 

III.  ADVANTAGES  OF  GRUBE'S  METHOD  .  .  .13 

IV.  DIRECTIONS  TO  TEACHERS.  .  .  .  .17 
V.  THE  FIRST  YEAR      ......      19 

First  Step — The  One     .  .  .  .  .19 

Second  Step — The  Two  .  .  .  .22 

Third  Step— The  Three  .  .  .      26 

Fourth  Step— The  Four  .  .  .  .30 

Fifth  Step— The  Five    .  .  .  .  •       34 

Sixth  Step— The  Six      .  .  .  .  -37 

Seventh  Step — The  Seven         .  .  .  .41 

Eighth  Step— The  Eight  .  .  .  •    '  44 

Ninth  Step— The  Nine  .  .  .  .  -47 

Tenth  Step— The  Ten  .  .  .  .  .50 

VI.  THE  SECOND  YEAR.    OBSERVATIONS  .  .  .54 

Eleventh  Step— The  Eleven      .  .  .  •       5$* 

Twelfth  Step— The  Twelve      .  .  .  .58 

Thirteenth  Step— The  Thirteen  .  .  .62 

Fourteenth  Step — The  Fourteen  .  .  .63 

Fifteenth  Step— The  Fifteen    .  .  .  .64 

Sixteenth  Step — The  Sixteen    .  .  .  .66 

Seventeenth  Step — The  Seventeen  .  .  .68 

Eighteenth  Step — The  Eighteen  .  .  .70 

Nineteenth  Step — The  Nineteen  .  .  .71 

Twentieth  Step — The  Twenty  .  .  -73 


CONTENTS. 


YI.  THE  SECOND  YEAR.    OBSERVATIONS.— Continued. 

Thirtieth  Step — The  Thirty      .            .            .  -77 

Fiftieth  Step— The  Fifty  .                     .            .  .80 

Hundredth  Step — The  Hundred           .            .  .82 

VII.  APPENDIX  TO  SECOND  YEAR'S  WORK      .           .  .90 

Section  I. — First  Unity              .            .            .  .90 

Second  Unity          .            .            .  •      94 

Section  II.— Third  Unity            .            .            .  .96 

Fourth  Unity          .            .            .  .98 

Fifth  Unity              .            .            .  -99 

Section  III.— Tenth  Unity           .            .            .  .     101 

Eleventh  Unity       ....     104 

Twelfth  Unity         .            .            .  .105 

Thirteenth  Unity    .            .            .  .106 

VIII.  SECOND  COURSE.    THE  THIRD  YEAR      .           .  .107 

I.  First  Half  of  the  Third  Year           .            .            .  .107 

A.  The  Pure  Number — First  Step            .           .  .108 

Second  Step        .            ,  .     109 

Third  Step          .            .  .     in 

Fourth  Step        .            .  .113 

Fifth  Step           .            ,  .115 

Sixth  Step           .            .  .118 

B.  The  Applied  Number               .           .           .  .118 

II.  Second  Half  of  the  Third  Year                .            .  .122 
"    A.  With  Abstract  Numbers— First  Step  :  Numeration  .     123 

Second  Step  :  Addition  .     127 

Third  Step  :  Subtraction  .     130 
Fourth  Step :  Multiplication    133 

Method  for  School  Work                    .           .  .     136 

Fifth  Step — Division  .     140 

Oral  Method— A.  Without  Remainder  .  .     140 

— B.  With  Remainder        .  .     142 

Written  Method— A.  Without  Remainder  .  .     143 

B.  With  Remainder         .  .     146 

Concrete  Numbers  ....     150 

IX.  THIRD  COURSE — FRACTIONS.    THE  FOURTH  YEAR  .    155 

I.  First  Half  of  the  Fourth  Year           .            .  .     155 

First  Step :  Halves               .            .            .  .156 


CONTENTS. 


IX.  THIRD  COURSE.    THE  FOURTH  YEAR.— Continued. 

Second  Step  :  Thirds  ....     159 

Third  Step :  Fourths  .  .  .  .162 

Fourth  Step:  Fifths  .  .  .  .166 

Fifth  Step :  Sixths  .  .  .  .169 

II.  Second  Half  of  the  Fourth  Year       .  .  .170 

1.  Unity1        ......     171 

2.  Expansion  and  Reduction  .  .  .172 

3.  Common  Denominator  .  .  .174 

4.  Number  relations  in  fractional  form      .  -175 


INTRODUCTION. 


PESTALOZZI  was  the  pioneer  who  broke  new  ground  in 
elementary  instruction  and  led  the  way  from  mechanical, 
abstract  methods  to  those  which  are  more  natural  and 
psychological.  He  laid  down  the  principle  that  all  mathe- 
matical-knowledge is  founded  upon  immediate  observation, 
and  therefore  must  proceed  from  the  concrete  to  the  general 
or  abstract  by  means  of  innumerable  examples.  This  dis- 
covery was  not  only  of  vast  importance  to  pupils  in  the 
schools,  but  it  opened  up  to  teachers  the  psychological 
principles  of  all  pedagogics. 

In  1842,  only  fifteen  years  after  Pestalozzi's  death,  ip- 
peared  Grube's  "  Leitfaden  fur  das  Rechnen  in  der  Ele- 
mentarschule."  In  this  work  Grube  gives  a  thoroughly 
developed  system  of  teaching  number.  Pestalozzi  was  un- 
fortunately lacking  in  system.  While  he  brought  to  light 
pedagogical  principles,  he  developed  no  system  of  peda- 
gogics. He  taught  the  world  that  the  proper  way  to  teach 
the  child  is  to  go  directly  to  Nature,  let  her  operate  on  the 
mind  and  follow  her  harmonious  development.  Grube 
found  the  germ  of  his  system  in  Pestalozzi's  teachings, 
but  went  farther  than  his  master  in  that  he  broke  away 
from  the  idea  of  teaching  the  four  processes,  addition,  sub- 
traction, multiplication,  and  division,  separately  and  in  the 
order  named.  This  is  one  of  the  great  and  most  impor- 
tant features  of  the  Grube  system.  Grube  held  that  the 


1 2  IN  TROD  UCTION. 

four  simple  processes  of  arithmetic  should  go  together  in 
the  small  numbers,  believing  it  to  be  the  natural  process 
of  the  mind. 

By  the  use  of  objects  the  child  is  brought  to  see  the 
relations  of  numbers  until  he  is  able  to  reproduce  the  re- 
lations without  the  objects.  As  the  elementary  work 
properly  covers  the  period  from  the  sixth  to  the  tenth 
year,  the  period  of  observation,  and  as  the  method  is 
purely  elementary,  Grube  discusses  only  the  first  four 
years'  work.  His  plan  provides  for  three  hours  (full 
hours)  per  week.  The  end  to  be  reached  is  a  thorough 
knowledge  of  the  fundamental  rules  and  common  frac- 
tions. His  work  is  divided  into  three  parts  or  courses : 
I.  Whole  numbers  from  i  to  100,  employing  the  first 

two  years. 

II.  Whole  numbers  above   100,  employing  the  third 
school  year. 

III.  Fractions,  employing  the  fourth  year. 

He  lays  down  the  work  definitely  for  each  half-year, 
which  we  shall  fully  discuss  later. 


ADVANTAGES  OF  GRLJBE'S  METHOD. 


I.  It  recognizes  the  psychological  fact  that 
nearly  all  the  knowledge  obtained  by  the  child 
in  its  earlier  years  is  by  means  of  the  senses. 

By  observation  and  not  by  reasoning  does  the  child  gain 
his  first  knowledge  of  number.  In  the  earlier  years  the 
child's  reasoning  powers  must  be  brought  very  little  into 
play.  He  is  not  yet  especially  ready  for  reasoning,  and 
Nature  did  not  intend  that  he  shall  gain  knowledge  at  this 
early  period  through  the  reason.  He  is  eager  for  knowl- 
edge, but  such  knowledge  as  is  obtainable  by  the  senses. 
He  learns  mechanically.  He  comes  to  know  all  the  com- 
binations and  manipulations  taught  him  so  as  to  give 
them  with  absolute  accuracy  and  great  rapidity  without 
stopping  to  think. 

II.  As  it  makes  the  first  year's  work  a  study 
of  the  numbers  1  to  1 0  it  lays  a  solid  foun- 
dation. 

The  knowledge  thus  obtained  becomes  an  acquirement 
which  will  be  a  methodical,  substantial  product.  Accord- 
ing to  our  usual  crude  methods  this  may  seem  but  little 
work  for  an  entire  year,  yet  by  this  scientific  study  the 
first  ten  numbers  will  be  found  to  furnish  ample  work. 
They  are  the  foundation  of  the  whole  number  system  ;  all 
larger  numbers  are  only  a  repetition,  in  a  sense,  of  the 
first  orders.  The  more  thoroughly  the  numbers  from  one 
to  ten  are  known,  the  surer  and  more  rapid  will  be  all 
later  work  in  arithmetic.  Let  this  foundation  be  well 


14        ADVANTAGES  OF  GRUBE'S  METHOD. 

laid  and  the  structure  is  well  begun.  Within  these  limits 
there  is  so  much  rich  material  for  all-sided  practical  ap- 
plications that  the  teacher  will  find  plenty  to  do  to  accom- 
plish the  teaching  of  the  first  ten  numbers  in  one  year. 

III.  The  Grube  method  progresses  gradually 
and  naturally  according  to  the  ability  of  the 
child. 

It  proceeds  from  the  knowledge  already  gained  to  new 
knowledge  by  a  very  easy  step.  The  knowledge  possessed 
is  utilized  in  mastering  new  knowledge.  The  child  must 
not  be  subjected  to  mental  over-exertion  at  any  penod. 
This  is  especially  dangerous  during  the  first  years.  The 
Grube  method  does  not  require  too  much,  and  yet  it  gives 
endless  and  suitable  variety  so  that  the  child  does  not  tire 
of  number. 

IV.  It  develops  the   mental  powers  evenly 
and  in  all  directions. 

One-sided  teaching  should  always  be  avoided.  All  de- 
velopment should  be  harmonious  and  natural.  Grube 
considers  each  number  by  itself  as  an  entirety  and  teaches 
all  about  it  completely,  thus  building  the  mathematical 
structure  step  by  step. 

V.  Elementary  teaching  of  number  should 
proceed  from  observation,  or,  better,  it  should 
proceed  from  things. 

Grube's  system  calls  for  the  use  of  things — balls,  marbles, 
cubes,  blocks.  It  uses  objects  repeatedly  until  the  child 
is  thorough  master  of  the  number  and  can  make  the  com- 
binations abstractly.  After  a  time  the  simple  remem- 
brance of  the  objects  used  will  always  be  sufficient  to  re- 
call to  the  consciousness  the  number  until  no  object  is 
longer  necessary  and  the  pure  relations  of  numbers  are 
fixed.  Then  the  child  needs  no  object,  or  intermediate 
process,  to  help  him  to  know  the  number,  but  he  knows 


ADVANTAGES  OF  GRUBE' $  METHOD.         Ij 

it  instantly,  and  the  simple,  fundamental  processes  are 
mechanical.  So  we  pass  from  the  object  to  the  symbol, 
from  this  to  the  comprehension  of  number,  and  lead  in 
this  way  the  interest  from  the  empire  of  objects  over  to 
the  empire  of  the  number  forms. 

VI.  The  Grube  method  makes  the  teaching 
of  number  an  excellent  language-lesson. 

The  answers  and  statements  are  to  be  complete  sen- 
tences ;  and  as  the  subject  is  always  kept  within  the  ability 
of  the  child  to  comprehend,  the  number-lesson  becomes 
one  of  the  most  valuable  means  of  teaching  language. 

VII.  The  child  acquires  the  habit  of  close 
observation. 

As  only  that  which  is  within  the  child's  comprehension 
is  brought  before  him,  and  as  familiar  objects  are  placed 
before  him  so  frequently  and  so  systematically,  he  acquires 
the  habit  of  accurate  and  close  observation.  He  learns 
to  see  what  is  brought  to  his  notice,  and  to  see  all  about 
it.  This  is  one  of  the  most  important  features  of  the 
Grube  method,  in  that  it  is  thoroughly  psychological. 

VIII.  It  develops  and  trains  the  attention. 

As  the  child  can  understand  the  matter,  it  interests  him, 
and  interest  is  the  first  factor  of  attention.  Because  of 
its  harmonious,  all-sided  development  it  cultivates  the 
power  of  attention  and  leads  the  child  to  the  habit  of 
commanding  and  fixing  the  attention  at  will. 

IX.  It  forms  the  habit  of  thoroughness  in 
the  child. 

Mastering  each  number  in  all  its  details  and  possible 
combinations,  it  becomes,  like  a  habit,  a  part  of  the  very 
being  of  the  child,  until  he  is  able  to  use  it  exactly  as 
he  uses  the  eye  or  the  hand,  without  conscious  thought. 


l6        ADVANTAGES  OF  GRUBE  S  METHOD. 

Beginning  thus  early  in  his  school  life  to  gain  a  thorough 
mastery  of  each  step,  he  is  led  to  desire  mastery  in  other 
departments  of  learning  and  of  practical  life. 

X.  The  Grube  method  gives   pleasure  and 
awakens  a  love  for  the  study  of  number. 

If  the  pleasure  of  work  is  not  found  in  the  work  itself, 
all  incentives  and  threatenings  will  be  in  vain.  The  desire 
to  know  a  thing  must  be  produced  in  the  child  himself. 
The  impulse  can  begin  in  the  young  mind  only  when 
there  is  the  consciousness  of  continual  unity  in  the  de- 
velopment of  his  powers,  and  then  he  will  be  driven  by 
this  impulse  to  further  development  by  his  self-activity. 
This  method  contains  such  unity  and  thus  awakens  in  the 
child  a  love  for  the  subject. 

XI.  It  makes  the  child  self-active  in  a  proper 
manner. 

Becoming  complete  master  of  a  number,  he  is  able  to 
combine  and  operate  with  it  by  making  original  examples. 
Thus  number  becomes  to  him  from  the  first  a  living, 
practical  reality. 

XII.  The  Grube  method  is  a  logical  one. 

It  proceeds  systematically  and  according  to  an  order  of 
sequence ;  it  is  psychological  in  that  it  teaches  the  use  of 
the  senses,  in  that  it  proceeds  from  the  simple  to  the 
more  difficult,  and  in  that  it  goes  out  from  the  known  to 
unknown  and  makes  constant  use  of  the  known ;  it  is 
practical  because  it  gives  a  sure  foundation  for  all  future 
work  in  arithmetic,  and  because  it  brings  the  child  im- 
mediately to  measure  and  compare  numbers  and  to  make 
use  of  the  relations  of  the  same. 


DIRECTIONS  TO' TEACHERS. 


THE  first  course  includes  whole  numbers  I  to  100.  Two 
years  are  required  for  this  work,  the  first  year  being  spent 
on  i  to  10  and  the  second  on  the  rest,  10  to  100. 

It  must  not  be  forgotten  that  the  number-lesson  must 
at  the  same  time  be  a  language-lesson.  It  is  of  the 
highest  importance  that  the  child  give  his  answers  in 
complete  sentences,  plainly  spoken,  with  clear  accent. 
Great  importance  must  be  attached  to  the  explanation  of 
every  example  from  the  outset.  So  long  as  the  child  is 
not  master  of  the  language  necessary  to  express  the 
operations  performed  with  the  number,  he  is  not  master 
of  the  representation  or  idea  of  the  number  itself,  he  does 
not  know  the  number. 

An  example  is  not  finished  when  the  answer  is  found, 
but  when  it  has  been  analyzed.  The  language  may  be 
taken  as  a  safe  test  that  a  pupil  has  completely  mastered 
a  step. 

So  far  as  possible  the  pupil  must  be  led  to  speak  for 
himself  and  not  to  depend  upon  half  the  answer  being  put 
into  his  mouth  by  the  teacher.  Concert  and  individual 
answers  must  interchange  in  order  that  the  interest  of  the 
class  be  maintained. 

The  uniform  objects  to  be  used  are  tbi  fingers  and 
blocks;  for  blackboard  or  slate  use  simple  straight  lines. 
Too  many  kinds  of  objects  must  not  be  used.  The  child 
has  only  a  certain  amount  of  strength  and  mental  power 
which  he  can  apply,  and  his  interest  must  not  be  divided. 

The  mental  comprehension  of  number  is  disturbed  if 
things  which  awaken  other  ideas  or  desires  are  employed. 
The  mind  is  capable  of  only  a  certain  amount  of  interest, 
and  when  this  interest  is  wholly  or  partly  withdrawn  but 
little  can  be  expected  for  the  particular  thing  at  hand. 


l8  DIRECTIONS   TO    TEACHERS. 

For  this  reason,  while  teaching  the  abstract  number,  there 
should  be  but  few  things  shown  the  child,  and  these 
should  be  simple  and  uniformly  the  same.  The  best 
things  are  blocks,  which  awaken  little  interest  in  them- 
selves, and  these  must  be  the  chief  objects  used  through- 
out. Other  objects  should  be  referred  to  after  the  child 
has  a  number  well  fixed,  byway  of  application,  but  should 
seldom  be  shown  him,  at  least  during  the  study  of  the 
first  numbers.  Thus  apples,  nuts,  etc.,  which  awaken 
desire,  stimulate  the  appetite,  and  thus  divide  the  atten- 
tion, must  not  be  used  as  objects  in  teaching  number.  All 
the  interest  which  the  child  gives  to  the  color,  taste,  etc., 
of  objects  is  just  so  much  lost  to  number. 

The  operations  in  any  step  consist  simply  of  compar- 
ing and  measuring  what  has  been  gained  in  the  preced- 
ing steps  with  that  which  is  new.  The  child  proceeds 
from  the  known  to  the  unknown,  from  the  easy  to  the 
difficult ;  hence  the  method  must  follow  a  psychological 
law.  Th3  pure  number  is  first  learned,  and  then  it  is  ap- 
plied to  things  in  order  to  fix  it  and  make  its  practical 
use  apparent. 

The  work  of  teaching  a  number  is  not  complete  until 
the  child  has  been  taught  to  make  neatly  and  with  dis- 
patch the  figure  which  stands  for  the  number.  This 
makes  excellent  employment  for  the  children  at  their 
seats,  and  is  a  good  preparation  for  written  arithmetic,  for 
which  the  foundation  is  being  laid. 

Go  slowly — clo  not  measure  the  ability  of  the  child  by 
your  ability;  bring  yourself  down  to  the  level  of  the 
child's  mind  ;  be  patient ;  repeat  everything  many  times ; 
review  daily;  use  many  examples  and  lead  the  children  to 
make  original  problems. 

Lastly,  do  not  expect  too  much  of  the  children ;  give 
them  the  kind  and  quantity  of  food  that  they  can  digest, 
remembering  that  real,  sound,  intellectual  growth  is 
slow,  especially  at  the  beginning. 


THE  FIRST  YEAR:  1-10. 


THE  work  of  the  first  year  embraces  a  study  of  the  num- 
bers, i,  2,  3,4,  5,  6,  7,  8,  9,  10.  Each  number  is  taken  by 
itself,  measured  by  those  that  precede  it,  compared  and 
studied  in  all  its  possible  operations. 


FIRST  STEP. 

THE  ONE.* 

The  one  can  only  be  measured  by  itself.     The  child  has 
only  to  learn  the  idea  of  unity. 


*  Many  insist  that  the  child  already  knows  the  one,  and  that  it  is  folly 
to  spend  time  in  teaching  it.  But  it  must  not  be  forgotten  that  the 
child  when  he  enters  school  must  begin  the  study  of  number,  must 
begin  to  form  habits  of  correct  speaking  and  thinking,  must  learn  to 
observe  carefully  what  is  done,  to  tell  what  he  sees  done,  and  to  answer 
in  complete  sentences.  Then,  too,  the  Grube  system  builds  step  by 
step,  always  making  use  of  the  knowledge  possessed.  The  one  is  the 
first  step.  For  these  reasons  and  for  these  purposes  a  short  time  can 
be  profitably  spent  with  the  one.  It  may  be  further  added  that  the 
fact  of  the  child's  knowing  the  one  will  make  it  especially  valuable  in 
getting  him  to  feel  confidence  in  himself,  an  important  matter  for  the 
child  when  he  begins  school. 


2O  THE  FIRST   YEAR:    i-io. 

I.  The  Pure  (Abstract)  Number.* 


(The  teacher  shows  the  block)   How  many  have  I  ? 

P.  You  have  one. 

T.  Charles,  take  this  block.  Now  how  many  has 
Charles  ? 

P.  Charles  has  one. 

T.  Charles,  give  it  to  me.  Now  how  many  has 
Charles? 

P.  Charles  has  none. 

T.  How  many  have  I  ? 

P.  You  have  one. 

(Lead  the  children  to  watch  the  movements  of  the 
teacher,  to  describe  accurately  what  he  does. 

Many  other  questions  similar  to  the  above  can  be 
given.  Great  care  must  be  given  to  the  language.  Cor- 

*  The  idea  of  a  number  is  given  the  child  by  the  use  of  an  object  or 
objects  which  are  placed  before  him  to  see.  He  sees  the  concrete  form, 
but  does  not  name  it.  It  suggests  to  him  the  abstract  number.  Thus 
when  the  child  sees  one  block  he  is  led  to  think  of  it  as  a  one,  two 
blocks  as  a  two,  three  blocks  as  a  three,  etc.  Do  not  let  the  child  say 
"  one  block,"  "  two  blocks,"  but  simply  u  one,"  u  two,"  etc.  Requiring 
the  child  to  name  the  object  withdraws  his  attention  from  the  number 
itself.  It  is  much  easier  for  him  to  speak  of  the  abstract  number,  and 
it  is  also  easier  for  him  to  read  and  write  it.  The  block  gives  him  the 
picture  of  the  number,  and  that  is  all  that  is  wanted  of  it ;  after  getting 
the  picture  of  it  he  only  needs  to  use  the  name  that  stands  for  the  num- 
ber. After  becoming  thoroughly  familiar  with  the  number,  the  child 
must  go  a  step  farther  and  use  it  with  various  objects  and  make  ap- 
plication of  it.  This  procedure  is  entirely  psychological  in  that  it  pro- 
ceeds from  the  simple  to  the  complex,  from  the  known  to  the  unknown. 


FIXST  STEP.  21 

rect  statements  must  be  given.  Never  accept  anything 
else.) 

T.  Now  let  us  put  this  one  on  the  board  (makes  a  draw- 
ing of  the  block).  How  many  have  I  ? 

P.  You  have  one. 

T.  But  it  takes  too  long  to  make  this  kind  of  a  one ;  we 
will  let  this  I  stand  for  our  one.  How  many  have  I  ? 

P.  You  have  I. 

(The  use  of  the  mark  is  simply  to  save  time.  The  child 
readily  proceeds  from  the  object  [the  block]  to  its  pic- 
ture, and  from  that  to  the  straight  mark  [  I  ]  as  standing 
for  one.) 

T.  Now  we  will  learn  to  make  something  else  that 
stands  for  one.  Watch  me  and  see  what  it  is.  It  is  this 
/.  What  does  that  stand  for  ? 

P.  That  stands  for  one. 

T.  Now  you  may  make  that  on  your  slates. 

(By  this  method  the  pupils  are  taught  the  figure.  The 
block,  its  picture,  and  the  mark  [I]  must  be  taught  as  the 
number  one.  The  child  thinks  of  them  only  as  the  one. 
But  he  must  learn  to  make  the  character  which  stands 
for  the  one  ;  namely,  the  figure  [/].  No  number  must  be 
left  and  no  step  be  considered  complete  until  the  pupils 
have  learned  to  make  the  figure  which  stands  for  the 
number.  Great  care  must  be  taken  by  the  teacher  not  to 
confuse  the  number  with  the  figure.) 

II.   The  Applied  Number, 

T.  What  thing  do  you  find  but  once  in  the  school- 
room? 

P.  I  find  one  stove  (desk,  clock,  etc.). 

T.  What  have  you  one  of  at  home  ? 

P.  I  have  one  dog  (cat,  sled,  etc.). 

(But  little  time  need  be  given  to  the  one,  the  aim  being 
chiefly  to  get  good  expressions  from  the  children,  and  to 
start  them  in  the  mode  of  thought  to  be  pursued  here- 
after with  the  other  numbers.) 


22  THE  FIRST   YEAR:    i-IO. 

SECOND  STEP. 

THE  TWO. 
I.  The  Pure  Number. — Measuring  and  Comparing. 


T.  How  many  have  I  ? 

P.  You  have  two. 

T.  ( Takes  a  block  in  each  hand  and  holds  up  one.)  What 
have  I  here  ? 

P.  You  have  one. 

T.  And  what  have  I  here  ?  (the  other  hand.) 

P.  You  have  also  one. 

T.  Now  watch  and  see  what  I  do.  (Moves  the  hands 
holding  the  blocks  slowly  together?)  What  did  I  do  ? 

P.  You  put  one  and  one  together. 

(Continue  this  operation  until  they  can  make  a  good 
statement  of  the  fact.) 

T.  Very  well.  Now  we  will  put  this  little  story  on  the 
board.  []  and  Q  make  p  Q  That  is  good  ;  but  I  can 
make  these  marks  [II]  instead  of  the  pictures  [Q  O], 
and  that  will  be  easier.  What  have  I  here  [I  I]  ? 

P.  You  have  two. 

T.  But  I  can  make  something  else  which  means  two. 
It  is  this  :  2.  What  have  I  now  ? 

P.  You  have  two. 

(The  teacher  should  in  this  manner  teach  each  figure. 
When  the  pupils  have  learned  the  number,  and  have  seen 
it  expressed  by  pictures  of  blocks  and  by  marks,  they  are 
then  ready  to  learn  the  simplest  way  of  expressing  it,  that 
is,  by  the  figure.  Make  the  figure  slowly  before  the  class, 
so  that  they  can  imitate  your  manner  of  making  it.) 


SECOND   STEP.  23 

T.  Now  what  have  I  ?     (Two  blocks  shown  together.) 

P.  You  have  a  two. 

(It  is  most  important  here  that  the  child  be  taught  that 
the  two  cubes  is  a  2  and  not  two  blocks.  The  child  must 
think  of  it  in  its  entirety — a  two.) 

T.  How  many  2's  have  I  ? 

P.  You  have  one  2. 

T.  How  many  times  have  I  2  ? 

P.  You  have  2  one  time  or  once. 

T.  How  many  does  one  2  make  ? 

P.  One  2  makes  2. 

T.  Now  let  me  write — One  2  is  2. 

(This  will  be  reached  only  after  repeated  and  patient 
efforts,  but  it  pays  to  give  the  necessary  time,  because  it 
is  fundamental  and  all  later  work  will  be  better  and  more 
easily  done.  Show  the  blocks  and  lead  the  child  to  see 
for  himself.  Always  go  back  to  the  blocks  when  the 
child  is  in  doubt.  After  the  child  has  learned  the  figure 
it  may  be  used  to  indicate  the  number.  Until  then  use 
objects,  pictures,  or  marks  to  stand  for  the  number.) 

T.  Again,  notice  what  I  have.  (The  two  blocks  in  one 
hand.) 

P.  You  have  2. 

T.  Now  see  what  I  do.    (  The  teacher  slowly  takes  i  from 

2.) 

P.  You  took  i  away  from  2. 

T.  And  what  does  that  leave  ? 

P.  It  leaves  i. 

T.  Who  will  tell  me  the  whole  story  about  what  I  did  ? 

P.  You  took  i  away  from  2  and  it  left  i. 

T.  Good ;  we  will  write  that  also. 

D  D  less  n  leaves  Q 

Very  well ;  now  we  will  write  that  another  way. 
2  less  i  leaves  i. 

Let  us  now  see  what  else  we  can  do.  (The  teacher  holds 
the  2  blocks  before  the  children  and  takes  i  away)  What 
have  I  done? 

P.  You  have  taken  i  away. 

(  Teacher  then  takes  the  other  away)  Now  what  have  I 
done  ? 


24  THE  FIRST   YEAR:  i-io. 

P.  You  have  taken  i  away  again. 

T.  How  many  times  have  I  taken  i  away  from  2  ? 

P.  You  have  taken  i  away  two  times. 

T.  Then  how  many  I's  are  there  in  2? 

P.  There  are  two  I's  in  2. 

T.  Now  we  will  write  that  on  the  board. 
In  2  there  are  two  i's. 

Or  we  may  say  2  divided  by  i  makes  2. 

(This  last  statement  will  be  somewhat  difficult  for  the 
child,  and  should  not  be  attempted  until  by  the  method 
above  given  he  has  mastered  the  idea.  Then  we  may 
teach  this  as  another  way  of  expressing  the  same  thing. 
It  is  necessary  that  the  child  learn  this  way  of  expressing 
division,  as  it  is  the  simplest  expression  and  is  the  form 
that  he  will  most  often  meet  with.  After  mastering  the 
idea  it  will  not  take  long  for  the  child  to  learn  the  form 
in  common  use.) 

T.  Now  we  will  write  what  we  have  learned  about  2. 

1  and  i  make  2. 

One  2  is  a  2  or  makes  a  2. 
Two  I's  are  a  2  or  make  a  2. 

2  less  i  is  i. 

In  a  2  there  are  two  i's.* 
In  a  2  there  is  one  2. 

(At  this  time  the  signs  can  be  taught.     Write  each  ex- 
pression using  the  words,  and  underneath  write  the  same 
expression  using  the  signs.)     For  example  : 
i  and  i  make  2, 

1  +  1  =  2. 

2  divided  by  i  makes  2. 
2  -*- 1  =  2. 

(Require  the  child  to  read  the  expression  containing  the 
words,  and  then  that  containing  the  signs.  He  must  read 
both  exactly  alike.  Hereafter  use  only  signs,  and  require 
the  pupils  to  make  and  use  them.  Go  slowly,  repeat 
many  times,  seek  to  get  correct  expressions,  teach  the 

*  I  can  take  i  from  2  twice. 


SECOND   STEP.  2$ 

children  to  observe  accurately  what  you  do  and  to  de- 
scribe it.) 

The  four  processes  must  be  repeated  until  the  pupils 
can  give  all  operations  with  great  rapidity.  They  must 
also  be  able  to  make  their  statements  fluently,  to  read 
readily  from  the  board,  and  to  write  exercises  from  dicta- 
tion, all  to  be  included  within  the  2. 

The  pupil  must  be  able  to  answer  such  combinations  as 
follows,  the  teacher  developing  them  by  use  of  the  blocks  : 

What  number  is  found  twice  in  2  ? 

Of  what  number  is  2  the  double  ? 

Of  what  number  is  i  the  half? 

What  number  must  I  double  in  order  to  get  2  ? 

I  know  a  number  which  has  i  more  than  i.  What  is 
it? 

What  number  must  I  add  to  i  in  order  to  get  2  ? 

All  possible  combinations  of  the  2  should  thus  be 
given. 

II.  The  Applied  Number. 

(The  pupils  are  now  prepared  to  apply  their  knowledge 
in  practical  examples  embracing  other  objects  than  the 
blocks.  It  is  no  longer  necessary  to  show  them  the  ob- 
jects. Let  them  also  make  examples.) 

Fred  had  2  cents  and  spent  i  cent  for  cherries.  How 
much  had  he  left  ? 

A  slate  pencil  cost  one  cent.  How  much  will  2  pencils 
cost  ? 

Charles  had  I  dime  in  his  savings  bank ;  his  sister  had 
twice  as  many.  How  many  had  his  sister  ? 

If  a  cake  cost  i  cent,  how  many  cakes  can  you  buy  for 
2  cents? 

James  had  2  apples  and  Frank  had  half  as  many.  How 
many  had  Frank  ? 

George  had  i  marble  and  John  twice  as  many.  How 
many  had  John  ? 


26  THE  FIRST  YEAR:  i-io. 

THIRD  STEP. 

THE    THREE. 


I.  The  Pure  Number. — Measuring  and  Comparing, 
(a)— With  one. 

(The  cubes  should  be  placed  on  a  table  where  all  the 
class  can  see  what  is  done  with  them.  It  is  inconvenient 
to  operate  with  three  or  more  blocks  in  the  hands.  There- 
fore, the  blocks  must  be  shown  on  the  table.) 

T.  How  many  have  I  ?     (Shows  the  3  blocks  separately.} 

P.  You  have  i  +  i  +  i. 

T.  How  many  is  that? 

P.   i+i  +  i  =  lll. 

T.  Good  ;  I  will  print  this  little  story  on  the  board  : 

p  +  D  +  Q = D  Q  O 

then 

i  +  i  +  i  =  lll. 

(When  sure  that  the  pupils  know  the  3,  he  writes 
1  +  1  +  1  =  3,  teaching  the  figure  3  by  the  same  method 
as  the  2  was  taught.  The  symbol  representing  the 
number  must  be  taught  as  soon  as  the  pupils  have  ob- 
tained the  idea  of  the  number,  and  not  before.  After  they 
have  learned  the  figure  which  stands  for  a  number,  no 
other  characters  need  be  used  to  represent  the  number.) 

T.  Now  tell  me  how  many  times  I  have  i.  (Picks  uptke 
blocks  one  after  another,  the  children  counting?) 

P,  You  have  one  3  times. 


THIRD   STEP.  27 

T.  One  3  times  makes  how  many? 
P.  One  3  times  makes  3. 
T.  Then  how  many  are  3  times  one  ? 
P.  3  times  i  are  3,  or 

3x1=3. 

T.  Now,  what  have  I  done  ?  (Holds  up  the  3  blocks  and 
takes  i  away.) 

P.  You  have  taken  i  away  from  3. 

T.  How  many  are  left  ? 

P.  There  are  2  left. 

T.  Who  will  write  that  for  me  on  the  board  ? 

P.  (Child  writes  3—1  =  2.) 

T.  What  else  have  I  done  ?  (Takes  another  block  away 
from  the  remaining  2.) 

P.  You  have  taken  i  away  from  the  2. 

T.  How  many  does  that  leave  ? 

P.  It  leaves  i. 

T.  Now  read  3  —  1  —  1  =  1. 

Still,  again,  how  many  times  can  I  take  I  away  from  3  ? 
( Takes  i  away  3  times,  children  counting?) 

P.  You  can  take  i  away  3  times. 

T.  Then  how  many  ones  are  there  in  3  ? 

P.  There  are  3  ones  in  3. 

T.  Then  I  will  write  i  in  3  =  3  or  3-7-1  =  3.  Now  see 
what  we  have  learned. 

,    fi  +  i  +  i=3 
3Xi         =3 

3-1-1=1 

U+;i        =3 

(b)— Measuring  with  2. 

(In  the  same  manner  the  following  tables  will  be  devel- 
oped, the  children  always  seeing  the  manipulations  with 
the  blocks  and  acquiring  the  statements  of  themselves. 
If  a  child  hesitates  and  does  not  yet  comprehend,  go  over 
the  operations  again.  Call  him  up  and  let  him  handle  the 
blocks  until  he  has  mastered  the  process  and  understands 
the  relations.) 


28  THE  FIRST  YEAR:  i-io. 

'2+1=  3,  1+2  =  3 
I  I    1  I  x  2  +  i  =3 

I     |3  —  2  =  i,  3—1=2 

3-7-2  =  1,  and  i  remainder.* 
3  is  I  more  than  what  ? 
3  is  i  more  than  2. 
3  is  2  more  than  what  ? 
3  is  2  more  than  I. 
2  is  i  less  than  what  ? 

2  is  i  less  than  3. 

In  the  same  manner  find  that — 
I  is  2  less  than  3. 

3  is  3  x  i. 
i  is  i  of  3.1 

What  3  equal  numbers  make  3  ? 

What  numbers  with  i  make  3  ? 

The  3  contains  a  2  and  a  i. 

(All  of  these  combinations  have  been  wrought  out  using 
the  blocks.  Use  only  the  blocks,  the  fingers  and  marks. 
Do  not  divide  the  attention  by  the  introduction  of  new 
objects  or  things  that  awaken  a  desire  of  possession  in 

*  This  should  be  developed  as  follows  : 

T.  How  many  have  I  here  ?    (Holding  up  the  3  blocks.) 

P.  You  have  3. 

T.  How  many  ones  in  it  ? 

P.  There  are  3  ones. 

T.  Now  see  how  many  twos  you  can  find.  (Children  take  the  blocks 
and  find  out  for  themselves.)  How  many  are  there  ? 

P.  There  is  one  2. 

T.  Who  will  find  another  2  ? 

P.  There  is  no  other  2,  there  is  only  i  left. 

T.  Then  how  many  twos  in  3  ? 

P.  There  is  one  2. 

T.  And  what  is  there  left  ? 

P.  There  is  i  left. 

T.  Now  tell  me  the  whole  story. 

P.  In  3  there  is  one  2  and  i  left. 

T.  Very  well,  but  I  like  the  word  remainder  instead  of  left.  Now 
try  again. 

P.  In  3  there  is  one  2  and  i  remainder. 

t  To  teach  this  I  would  take  3  blocks  and  place  i  block  near  them. 
Then  ask  which  is  the  larger  and  how  many  times.  Reverse  the  pro- 
cess, and  ask  which  is  the  smaller,  how  many  of  the  ones  it  takes  to 
make  a  three,  and  finally,  what  part  of  the  three  the  one  is. 


'THIRD   STEP.  29 

the  child.     Require  the  pupils  to  make  the  figures  I,  2,  3 
forwards  and  backwards,  thus,  i,  2,  3,  3,  2,  i.) 

Rapid  Work. 

Teacher  gives  examples  orally  as  rapidly  as  possible, 
the  children  giving  only  the  answer. 
How  many  are  3—1  —  1  +  1? 
3x1—2x1—1=? 
1+1x1+1—3=? 

3—2+1+1—2=? 

2  +   1+1—2  +   1  —  1=? 

(Many  examples  of  this  kind  should  be  given  until  the 
pupils  are  able  to  give  the  answer  instantly.  This  is 
largely  oral  work.) 

Combining. 

From  what  number  can  you  take  the  double  of  i  and 
still  have  i  remaining  ? 

What  number  is  3  times  i  ? 

I  use  a  number  once,  and  then  once  again,  and  then 
once  again,  and  obtain  3.  What  is  the  number? 

n.   The  Applied  Number. 

If  you  would  buy  a  3-cent  stamp,  how  many  cents  must 
you  have? 

Anna  had  a  3-cent  piece,  and  bought  2  cents'  worth  of 
candy.  How  much  change  should  she  get  ? 

If  a  pencil  cost  i  cent,  how  much  will  3  pencils  cost? 

Charles  has  2  apples  and  i  apple.  How  many  apples 
has  he  ? 

Mary  divided  3  flowers  among  her  father,  mother,  and 
brother.  How  did  she  divide  them  ? 

Martha,  Fanny,  and  William  have  each  i  book.  How 
many  books  have  they  ? 

A  boy  had  3  apples  and  ate  i  apple.  How  many  had 
he  left? 

David  has  i  dollar.  How  much  more  must  he  earn  in 
order  to  have  3  dollars  ? 

An  uncle  divided  3  dimes  equally  among  his  3  nieces. 
How  much  did  each  receive? 

My  father  gave  each  of  his  boys  i  dollar,  and  it  took  3 
dollars.  How  many  boys  had  he  ? 


3O  THE  FIRST  YEAR:  i-io. 

FOURTH  STEP. 

THE    FOUR. 


(It  will  not  be  necessary  longer  to  pursue  the  question 
and  answer  method,  as  it  has  been  sufficiently  illustrated. 
The  course  indicated  should  be  pursued  for  all  subsequent 
numbers.  Develop  all  the  relations  which  come  under 
"  Measuring  and  Comparing"  by  use  of  the  blocks.  Lead 
the  children  to  the  statement  of  what  you  develop.  The 
further  you  proceed  the  greater  opportunity  for  variety, 
but  limit  that  to  such  as  may  be  obtained  by  use  of  such 
objects  as  have  been  heretofore  specified,  namely,  blocks, 
fingers,  marks.) 

I.  The  Pure  Number. — Measuring  and  Comparing. 

(a)  With  i. 

i  i  i  i  4 

i    /   r       1  +  1  +  1  +  1=4, 1  +  1  =  2  +  2  =  4. 
i    t  \       4x1=4. 

|     ;  4_!  _i-i  =  i. 

I     /    [         4  +  i  =  4. 

(b)  With  2. 

r      2  +  2  =  4. 

I     I     2   J          2x2=4. 
I     I     2    1          4—2  =  2. 

[  4+2  =  2. 


FOURTH  STEP.  31 

(c)  With  3. 

f         3  +  i  =  4,  i  +  3  =  4- 
Ill  1x3  +  1=4. 

I  4  — 3  =  i,  4— 1=3. 

4-5-3=1  and  i  remainder  (3  in  4  once, 

and  i  remainder.) 

How  many  more  legs  has  a  horse  than  a  man  ? 
How  many  times  the  number  of  wheels  of  a  bicycle  has 
a  wagon  ? 

How  many  more  legs  has  a  chair  than  a  piano-stool  ? 
4  is  how  many  more  than  3  ? 
(Bring  out  the  following  facts  with  the  blocks): 
4  is  i  more  than  3,  2  more  than  2,  3  more  than  I. 

3  is  i  less  than  4,  i  more  than  2,  2  more  than  i. 
2  is  2  less  than  4,  i  less  than  3,  i  more  than  i. 

i  is  3  less  than  4,  2  less  than  3,  i  less  than  2. 

4  is  4  times  i,  2  times  (or  double)  2. 

i  is  one  fourth  of  4,  2  is  one  half  of  4. 

Of  what  equal  and  what  unequal  numbers  is  4  made  up  ? 

Rapid  Work* 

2x2  —  3  +  2X  i  +  i  —  2  doubled  =  ? 
4—  i  —  i  —  i  +  i  —  3  is  how  much  less  than  4 ? 
3—1  X2  —  3  +  2+1-1-2=? 

2X2X    I—  3  +  2—  I    X    2  -^  3  =  ? 

1+1+1—2x4—3x2=? 
4-^-2  +  1  +  1  —  3  —  1=? 

(Continue  the  work  as  described  before,  always  keeping 
within  the  combinations  of  the  4.) 

Combining. 

What  number  must  I  take  2  times  in  order  to  get  4  ? 
Of  what  number  is  4  the  double  ? 

*  These  expressions  are  intended  only  for  dictation,  the  pupil  work- 
ing as  fast  as  they  are  dictated,  and  obtaining  the  result  of  each  step 
with  no  reference  to  what  is  to  follow.  Thus  3  —  1x2  —  3  +  2+1-*- 

2  would    be  when  worked   out   from  dictation  3—  1  =  2,  2x2  =  4,  4  — 

3  =  1,    1+2  =  3,  3  +  1=4,  4-+-2  =  2.     Of    course    the    pupils    do   not  re- 
peat the  numbers  as  in  the  exercises  here  given,  but  obtain  each  result 
mentally  as  soon  as  the   teacher   dictates.    The  end  sought  is  rapidity 
as   well    as    accuracy.     When    expfessions   are"  written    for   the    pupils, 


32  THE  FIRST  YEAR:   i-io. 

Of  what  number  is  2  the  half  ? 

Of  what  number  is  i  the  fourth  ? 

What  number  can  be  taken  twice  away  from  4  ? 

What  number  is  3  greater  than  i  ? 

How  much  must  I  add  to  the  half  of  4  in  order  to  get  4  ? 

How  many  times  i  is  the  half  of  4  less  than  3  ? 

If  I  take  i  from  4,  how  many  times  i  have  I  left? 

If  I  add  i  to  i,  what  part  of  4  have  I  ? 

If  I  take  3  from  4,  what  part  of  3  have  I  left  ? 

How  much  is  the  half  of  4  more  than  the  third  of  3  ? 

How  much  is  the  fourth  of  4  less  than  the  half  of  4? 

II.  The  Applied  Number. 

Caroline  hajd  4  tulips  in  her  vase  which  she  neglected 
to  water.  One  wilted,  then  another,  then  another.  How 
many  had  she  left  ? 

How  many  cents  in  2  two-cent  pieces  ? 

In  a  one-cent  and  a  three-cent  piece  ? 

How  many  cakes  can  you  buy  for  4  cents  if  each  costs 
i  cent? 

When  each  costs  2  cents  ? 

If  a  top  costs  2  cents,  how  much  will  2  tops  cost? 

John  paid  for  2  cakes  a  three-cent  and  a  one  cent  piece. 
What  was  the  cost  of  each  ? 

One  quart  has  2  pints.     How  many  pints  in  2  quarts  ? 

Charles  had  4  chestnuts,  and  gave  Frank  i  and  Henry 
i.  How  many  had  he  left  ? 

William  had  3  peaches  and  ate  2.  How  many  had  he 
left? 

Anna  received  an  orange  on  Monday,  one  on  Tuesday, 
one  on  Wednesday,  and  one  on  Thursday.  How  many 
did  she  receive  in  all  ? 

care  must  be  taken  to  have  them  mathematically  correct  and  not  to  in- 
clude combinations  beyond  the  4,  or  the  number  which  is  being  taught. 
The  above  expressions,  as  well  as  those  which  follow  later  under  the 
head  of  "Rapid  Work,"  may  not  always  be  correct  when  taken  as  a 
whole  and  considered  as  a  mathematical  expression;  but  they  are  cor- 
rect as  dictation  exercises,  and  a  few  are  given  for  the  purpose  of  sug 
gesting  the  method  to  the  teacher.  Such  exercises  will  be  found  very 
valuable. 


FOURTH  STEP.  33 

George  had  4  apples  and  ate  i  each  day.  How  many 
days  did  they  last  ? 

If  a  pint  of  milk  costs  2  cents,  how  many  pints  will  4 
cents  buy  ? 

What  part  of  4  cents  is  i  cent  ? 

What  part  of  4  cents  is  2  cents  ? 

What  part  of  i  quart  is  i  pint  ? 

What  part  of  i  gallon  is  i  quart  ? 

My  father  had  3  cows  and  bought  i  more.  How  many 
has  he  now? 

If  he  sells  2  cows  how  many  will  he  have  left  ? 

Mary  had  i  pin  and  found  3  more.  How  many  had 
she  then  ? 

If  a  letter  requires  2  two-cent  stamps,  what  will  it  cost 
to  mail  it? 

I  had  4  quarts  of  milk,  and  sold  3  quarts.  How 
many  quarts  had  I  left  ? 

A  mother  has  2  sons  and  2  daughters.  How  many 
children  has  she? 

I  have  4  pears  which  I  wish  to  divide  equally  between 
my  2  sisters.  How  many  can  I  give  each  ? 

(Give  many  examples  from  the  life  of  the  children  until 
they  can  make  application  of  all  the  relations  contained 
within  the  4.  Let  them  also  make  examples.  Do  not 
let  them  leave  a  number  for  a  new  step  until  they  can 
make  the  figure,  perform  rapidly  all  the  combinations 
whether  given  to  them  orally,  or  written  on  the  board, 
make  complete  statements  of  such  combinations,  and 
apply  them  in  their  own  surroundings.  The  great  suc- 
cess of  this  method  depends  upon  thoroughness  in  these 
particulars.) 


34 


THE  FIRST  YEAR:   i-io. 


FIFTH  STEP. 

THE   FIVE. 


(As  it  is  difficult  to  work  with  five  blocks  in  the  hands, 
they  should  be  manipulated  on  a  table  before  the  children 
so  that  they  can  be  plainly  seen.  It  would  be  better  still 
if  the  children  could  be  gathered  around  a  large  table  and 
each  have  the  same  number  of  blocks,  so  as  to  perform 
the  same  work  as  the  teacher  does  with  hers.  The  blocks 
can  very  well  be  dropped  and  only  the  fingers  used  after 
this.) 

I.  The  Pure  Number. — Measuring. 

(a)  With  i. 

'     +  1  +  1  +  1  +  1=5 
5  x  i  =  5,  i  x  5  =  5 
5  —  i  —  i  —  i  —  i  =  i 


FIFTH  STEP.  35 

(b)  With  2. 

oo     2    f2  +  2  +  i  =5 
oo     2   \  2  X  2  +  '  =  5 

0/1   5-2-2  =  1 
[5  +  2  =  2(1) 

(<:)  With  3. 

r3  +  2  =  5,  2  +  3  =  5 
ooo    311x3+2=5 
oo        215-3  =  2,  5-2  =  3 

I  S  •*•  3  =  i  C2) 
(<0  With  4. 

f4  +  i  =  5,  i  +4  =  5 
oooo    4   J  i  x  4  +  i  =  5 
o  /    ]  5-4=  i,  5  -  i  =4 

[5+4=1  (i) 

After  the  pupils  are  familiar  with  all  the  combinations 
they  may  be  required  to  fill  out  missing  numbers  ;  for  ex- 
ample :3  + =5, 4x1  + =5,  4  x +1  =  5. 

5  is  i  more  than  4,  2  more  than  3,  3  more  than  2,  2 
more  than  i. 

4  is  i  less  than  5,  i  more  than  3,  2  more  than  2,  i  more 
than  3. 

3  is  2  less  than  5,  i  more  than  2,  2  more  than  I. 
2  is  3  less  than  5,  i  more  than  i. 

5  =  5x1. 

i  =  |  x  5  (i  is  the  fifth  part  of  5.) 
The  five  consists  of  unlike  numbers,  3  +  2,  and  of  2  like 
numbers  and  i  unlike  number,  2  +  2  +  1. 

Rapid  Work. 
5  —  2  —  2  +  2—1  X2,  the  half,  less  i  = 

2X2  +  1—  3  X   I   X2  —  3— 4  =  ? 

Combining. 

How  much  must  I  add  to  2  in  order  to  get  5  ? 

How  much  must  I  take  away  from  5  in  order  to  get  2  ? 

What  number  is  the  fifth  part  of  5  ? 

How  many  times  2  have  I  added  to  i  to  get  5  ? 


36  THE  FIRST  YEAR:    i-io. 

I  have  taken  from  a  number  twice  2  and  have  i  left 
What  is  the  number  ? 

If  to  2  times  2  I  add  i,  what  do  I  get  ? 

I  take  3  times  i  from  a  number  and  have  2  left.  What 
is  the  number? 

What  number  shall  I  add  to  2  to  get  5  ? 

If  I  add  i  to  a  number  I  get  5.    What  is  the  number? 

What  must  I  add  to  the  half  of  4  to  get  5  ? 

Take  2  times  i,  then  add  it  to  2 ;  this  lacks  how  many 
of  5? 

4  times  one  third  of  3  added  to  i  makes  what  ? 

Take  3  from  5  and  how  many  will  2  times  this  lack  of  5  ? 

(These  combinations  must  be  multiplied  to  insure  readi- 
ness and  accuracy.  While  in  the  examples  above  only 
those  involving  5  are  given,  review  of  the  numbers  already 
learned  must  never  be  forgotten.) 

II.  The  Applied  Number. 

How  many  3-cent  and  2-cent  loaves  of  bread  can  you 
buy  for  5  cents  ? 

John  received  from  his  father  a  3-cent  and  a  2-cent 
piece.  He  bought  2  sheets  of  paper  at  2  cents  each.  How 
much  change  did  he  get? 

Bertha  knit  3  times  around,  and  her  sister  2  times 
more  than  she.  How  many  times  around  did  the  sister 
knit? 

A  father  divided  5  cherries  among  his  3  children.  The 
youngest  got  only  i,  and  the  other  two  each  the  same 
number.  How  many  did  the  others  get? 

Charles  gave  his  2  sisters  each  i  apple,  his  brother  2 
apples,  and  had  i  for  himself.  How  many  did  he  have  at 
first? 

The  milkman  had  5  quarts  of  milk,  and  sold  2  quarts  to 
Mrs.  Wilson,  i  quart  to  Mrs.  Rand,  and  I  quart  to  me. 
How  many  quarts  did  he  have  left? 

John  has  2  marbles,  David  has  3.   How  many  have  both  ? 

John  wins  2  from  David.  How  many  has  David  left  ? 
How  many  has  John  now  ? 

How  many  apples  can  you  buy  for  5  cents  if  i  apple 
costs  you  2  cents  ?  How  many  cents  will  you  have  left  ? 


SIXTH  STEP, 


37 


(These  examples  must  be  multiplied,  making  use  of 
things  with  which  the  children  are  familiar.  Continue 
this  practice  until  they  can  perform  all  the  operations 
with  absolute  accuracy  and  great  rapidity.  Whenever  a 
child  is  in  doubt,  take  him  to  the  blocks  and  make  it  clear 
to  him.  Or,  better,  lead  him  to  find  out  the  truth  himself 
by  use  of  the  blocks.) 


SIXTH  STEP. 

THE  SIX. 


(The  pupil  is  now  able  to  fill  out  the  operations  himself 
according  to  the  method  already  pursued.  If  he  can  do 
that  readily,  it  may  not  be  necessary  to  use  the  objects 
longer,  except  when  the  child  is  puzzled.  Many  Germans 
use  no  objects  in  teaching  number  after  the  four.  Just 
as  soon  as  the  child  is  familiar  with  the  method,  and  can 
grasp  the  idea  of  the  number  without  the  visible  object 
before  him,  the  objects  should  be  abandoned.  But 


THE  FIRST  YEAR:     i-io. 


blocks  should  always  be  at  hand  to  be  used  in  removing 
doubt  in  the  child's  mind,  when  a  point  is  not  perfectly 
clear  to  him.) 


I.  The  Pure  Number.— Measuring. 


1+1+1+1+1+1=6 
6  x  1=6,  i  x6  =  6 
6— i— i— i— 1—1=1 
6-i-i  =6 


(Teach  each  figure  as  soon  as  the  child  has  use  for  it.) 


oo    2 
o  o    2 
oo    2 

1Z,  -f   «5  f  -^  =  <->.        ^    f   >! 
3x2  =  6 
6—2—2=2 
f 

=  4»  4  1 

6  +  2  =  3 

ooo    3 

[3  +  3  =  6 
1  2  x  3  =  6 

ooo    3 

|673=3 

[4  +  2  =  6,  2  +  4  =  6 

oooo    4 

1  i  x  4  +  2  =  6 

oo    2 

16-4=2 

[6^-4=1  (2) 

[5  +  1=6,  1  +  5=6 

ooooo    5 

I  i  x  5  +  i  =6 

0      / 

l6-5=i 

U-5  =  i(i) 

6  = 

5  +  i,  4  +  2,  3  +  3,  2  + 

4,     +  5 

5  = 

6—  i,  4  +  i,  3  +  2,  2  + 

3.     +4 

4  = 

6  —  2,  5  —  i,  3  +  i,  2  + 

2,     +3 

3  = 

6  —  3,  5  —  2,  4  —  i,  2  + 

I,       +2 

2  = 

6  -  4,  5  -  3,  4  -  2,  3  - 

I,       +  I 

I  = 

6  -  5»  5  ~  4,  4  -  3>  3  - 

2,  2  —  I 

6  = 

6xi,  3x2,  2x3 

3  = 

i  x  6  (is  half  of  6) 

i  x  6 

i  = 

i  x  6. 

SIXTH  STEP. 


39 


Of  what  3  like  numbers  is  6  composed  ?    Of  what  3 
unlike  ? 
The  following  tables  can  be  profitably  given  : 


(*) 


6  -  i  =  5 
6  —  2  =  4 

6~3  =  3 
6  —  4  =  2 
6-5  =  1 
6-6  =  0 

5  -  2  =  3 
3-2  =  1 


4  =  6 

3  +  3  =  6 

5  +  i=6 
(c)  1+2  =  3 

Odd  numbers  :  i,  3  5 ;     5,  3, 
2  +  2  =  4 

4  +  2  =  6 

Even  numbers :  2,  4,  6 ;  6,  4,  2. 
+  1=2 
+  2  =  3 
+  3  =  4 
+  4  =  5 
+  5  =  6 

(g)  6  =  6 

6  =  5  +  1 
6  =  4  +  2 

6  =  3  +  3 
6  =  2  +  4 
6=1+5 

(Many  other  tables  can  be  made  embracing  multiplica- 
tion and  division  as  far  as  the  6.  The  pupils  can  easily  be 
led  to  make  these  themselves.  Dictate  numbers,  and 
require  the  pupils  to  name  or  write  them  promptly,  so  as 
to  test  their  knowledge  of  the  order  of  the  numbers,  and 
their  ability  to  make  the  figures.  For  example  :  Write 
the  numbers  from  j  to  6.  What  number  comes  after  4? 


6-: 

4~- 

2  =  4 

I  =  2 

2  —  : 

I  =  0 

2. 

(/) 

6- 

=  5 

5- 

=  4 

4  — 

=  3 

3- 

—  2 

2  — 

=  I 

I  — 

=  0. 

w 

6- 

i  =  5 

6-: 

2  =  4 

6- 

5  =  3 

40  THE  FIRST  YEAR:     i-io. 

What  comes  before  4?  What  comes  between  2  and  4? 
Express  the  number  that  comes  after  5.  Express  the 
number  between  3  and  5.) 

Rapid  Work. 

i  x  2  +  i  x  2  —  i  x  1  —  5  +  5  =  ? 

4  +  2  —  3  is  how  much  less  than  6  ? 

3  —  2  x  5  +  i  —  4  x  2-7-2  +  4—5  =  ? 

5  —  4  +  3-5-  2  +  i  x  2-*-3  =  ?    « 

These  should  be  given  orally  to  the  pupils,  or  written 
on  the  board  as  rapidly  as  they  are  able  to  work  them. 
Allow  no  counting  of  the  fingers  or  use  of  objects  in  this 
operation.  The  pupils  must  know  eveiy  operation,  and 
be  able  to  perform  rapidly  and  accurately  without  any 
hesitation.  This  is  a  test  of  the  thoroughness  of  the  work. 

Combining. 

What  number  can  you  take  3  times  from  6  and  twice 
from  4? 

How  many  times  i  has  half  of  6  more  than  half  of  4, 
and  how  much  less  than  5  ? 

I  have  taken  a  number  twice  away  from  6  and  have  2 
left.  What  is  the  number? 

How  many  times  is  i  of  6  contained  in  4  ?  The  half  of 
4  =  what  part  of  6  ?  What  number  is  3  times  2  ? 

II.  The  Applied  Number. 

How  many  times  i  cent,  2  cents,  and  3  cents  in  6  cents  ? 

How  many  quarts  in  6  pints  ? 

What  will  3  liters  of  milk  cost  at  2  cents  a  liter  ? 

William  got  3  tops  for  6  cents.     What  did  i  cost? 

I  have  6  apples  in  3  pockets.  How  many  apples  in  each 
pocket  ? 

How  many  lead-pencils  at  2  cents  each  can  I  buy  for 
6  cents  ? 

I  gave  each  of  my  3  sisters  2  oranges.  How  many 
oranges  did  I  give  away  ? 

A  father  divided  6  dollars  equally  among  his  3  children. 
How  much  did  each  get  ? 


SEVENTH  STEP.  4! 

Fanny  took  6  cents  to  the  store  and  bought  2  candies 
at  2  cents  each.  How  much  money  did  she  have  left  ? 

Joseph  gave  Charles  2  marbles,  Henry  i,  and  had  3  left. 
How  many  had  he  at  first  ? 

I  have  5  dollars  and  borrow  i  more,  and  lend  2.  How 
much  have  I  left  ? 

If  David  earns  2  dimes  a  day,  how  much  will  he  earn 
in  3  days  ? 

Mary  gave  each  of  her  5  friends  a  candy,  and  had  i  for 
herself.  How  much  did  she  have  at  first  ? 

I  have  2  books  on  the  table,  i  on  the  chair,  and  3  in  my 
book-case.  How  many  have  I  in  all  ? 

Jessie  buys  3  pints  of  milk  at  2  cents  a  pint.  How 
much  is  the  cost  ? 


SEVENTH  STEP. 

THE    SEVEN. 


1+1+1+1+1+1+1=7 

7x1=7 

7— i— i— i— i— i— 1=1 

7-7-1=7 


42  THE  FIRST  YEAR:    i-io. 

00   2   f   2+2+2+1=7 

oo  2  I   3x2+1=7 

OO   2  ]   7—2—2—2=1 
O   1   (   7-H2  =  3(l) 

o  o  o  3  f  3  +  3+  1=7 
ooo  I  ^3  +  r=7 
oo.o  ,\  >+l-lul 

f  4  +  3  =  71  3  +  4  =  7 

oooo  4  1x4+3=7 

ooo  3  '  7  —  4  =  3 

[  7  +  4  =  i  (3) 

(5  +  2  =  7;      2  +  5  =  7 
i  x  5  +  2  =  7 
7  —  5  =  2 
7  -  5  =  i  (2)  * 

ooooootf    f     6  +  1=7;     1+6  =  7 
o/          1x6+1=7 
1     7-6=1 
(     7  -v-  6  =  i  (i) 

In  what  ways  can  a  father  divide  7  apples  among  2,  3,  4 
children  ? 

7  =  6  +  i,  5  +  2,  4  +  3,  3  +  4,  2  +  5,  i  +  6. 

6  =  7  -  i,  5  +  i,  4  +  2,  3  +  3,  2  +  4,  i  +  5. 
5  =  7  —  2,  6  —  i,  4  +  i,  3  +  2,  2  +  3,  i  +  4. 
4  =  7  —  3,  6  —  2,  5  —  i,  3  +  i,  2  +  2,  i  +3. 

7  =  7  x  i,  i  =  |  of  7. 

What  like  numbers  does  7  contain  ? 

(Make  tables  like  those  on  page  39,  and  require  the 
pupils  to  do  the  same.  Follow  this  course  with  all  suc- 
ceeding steps.) 

*  I  would  have  the  pupils  find  out  how  many  fives  in  7  by  use  of 
blocks.  Then  express  7-1-5  =  1  with  a  remainder  of  2.  Then  write 
the  remainder  in  parentheses,  as  :  7  •*-  5  =  i  (2).  The  children  will 
very  easily  learn  that  the  number  in  parentheses  is  the  remainder. 


SEVENTH  STEP.  43 

Rapid  Work. 
3x2  +  1  —  2X  1  —  3x3  +  1? 

2+I+2+I   +  I?     1+2+4  —  3  —  2X3? 


Combining. 

From  what  number  can  you  take  i  seven  times  ? 
What  number  contains  7  seven  times  ? 
To  what  number  must  I  add  3  x  2  to  get  7  ? 
I  take  a  number  3  times  and  get  i  less  than  7.    What 
is  the  number? 
How  many  times  i  is  7  greater  than  the  double  of  2  ? 

(The  double  of  2  is  4.  7  is  3  more  than  4,  and  has 
therefore  3x1  more  than  4.  Therefore  7  is  3  x  i 
greater  than  the  double  of  2.) 

II.    The  Applied  Number. 

A  week  has  seven  days.  What  is  the  name  of  the 
first,  the  second,  the  fifth,  the  third,  the  seventh  day? 

I  took  a  trip  lately  that  lasted  just  a  week  ;  how  many 
days  was  I  on  the  journey  ? 

How  much  money  did  I  need  for  the  journey,  if  I  used 
one  dollar  a  day  ? 

If  you  put  i  cent  in  your  savings  bank  each  day,  how 
much  will  that  make  in  a  week  ? 

How  many  threes  would  that  make? 

How  many  quarts  in  7  pints  ? 

George  was  sent  by  his  mother  to  fetch  2  3-cent  loaves 
of  bread.  She  gave  him  7  cents.  Was  that  enough  ? 
How  much  did  he  have  left  ? 

Henry  took  a  5-cent  and  a  2-cent  piece  and  bought  3 
candles  at  2  cents  each.  How  much  money  should  he 
bring  back  ? 

(The  teacher  must  multiply  examples  of  each  kind  until 
the  pupils  have  mastered  the  number.  Never  leave  a 
number  to  take  up  a  new  one  until  the  former  is 
thoroughly  learned.) 


44 


THE  FIRST  YEAR:  i-io. 


EIGHTH  STEP. 

THE    EIGHT. 


I.  The  Pure  Number. — Measuring. 


O  7 

O  / 

O  / 

O  / 

o  / 

o  / 

o  / 

o  / 

o  o  2 

o  o  2 

o  o  2 

o  o  2 


1  +  1  +  1  +  1  +  1  +  1  +  1  +  1=8 

8xi=8 

8—1—1—1—1—1—1—1=1 


2  +  2  +  2  +  2  =  8 

4X2  =  8 

8  —  2  —  2  —  2  =  2 

8  -f-  2  =  4 


0 
0 

o 
o 

0 

0 

o 

0 

3  f 

3 
M 

3 

2 

8 
8 

x 

3 
3 
3 
3 

+  2 
+  2 

-3 

=  2 

=  8 
=  8 

=  2 

(2) 

4 

+ 

4 

=  8 

0 

o 

o 

2 

X 

4 

=  8 

o 

o 

0 

8 

— 

4 

=  4 

8 

-r- 

4 

=  2 

EIGHTH  STEP.  45 


00000 

o  o  o 

.1 

3i 

5  +  3  =  8,  3  + 

1x5+3=8 

8-5  =  3 
8  -*-  5  =  i  (3) 

5=8 

( 

6  +  2  =  8,  2  + 

6  =  8 

000000 

6 

i  x  6  +  2  =  8 

O  0 

2 

8-6  =  2 

( 

8  •*-  6  =  i  (2) 

( 

7  +  i  =  8,  i  + 

7  =  8 

O  0  0  0  0  0  0 

7 

1x7+1=8 

0 

/ 

8-7  =  1 

8  •+-  7  =  i  (0 

8  =  7  +  i,  6  +  2,  5  +  3,  4  +  4,  etc. 

7  =  8  —  i,  6  +  i,  5  +  2,  4  +  3,  etc. 
6  =  8  —  2,  5  +  i,  4  +  2,  3  +  3,  etc. 

5  =  8  —  3,  etc.,  completing  the  table. 

8  =  2  x  4,  4  x  2,  8  x  i. 
i=ix8,  2=Jx8,  4  =  ^x8. 

The  8  consists  of  4  equal  numbers,  for  it  equals  4x2, 
and  of  2  equal  numbers,  for  it  equals  2x4,  also  of  2  equal 
and  one  unequal  number,  namely,  2x3  +  2. 

Rapid  Work. 
8  —  i  —  2  —  i  —  2  ? 

I+2  +  I+2  +  2  —  5? 
2X2  +  3  +  1-^-4  +  2? 

4  +  3-5  x  4-5-2-3  x  7? 
Combining. 

What  number  contains  the  fourth  of  8  three  times  ? 

What  is  the  difference  between  a  half  of  8  and  a  half 
of  6? 

What  number  must  I  double  in  order  to  get  8? 

What  number  must  I  take  4  times  in  order  to  get  8  ? 

What  number  has  5x1  more  than  3  ? 

(The  number  which  has  5  x  i,  or  5,  more  than  3,  is 
5  +  3  =  80 

Take  the  third  of  six  4  times. 


46  THE  FIRST  YEAR:  i-io. 


II.   The  Applied  Number, 

How  many  twos,  threes,  and  fours  are  found  in  8  ? 

How  many  gallons  in  8  quarts  ? 

How  many  quarts  in  8  pints  ? 

How  many  weeks  in  8  days  ? 

William  wanted  to  buy  4  spools  of  thread  at  2  cents 
each.  How  much  money  must  he  have?  He  paid  the 
sum  in  2-cent  pieces.  How  many  did  it  take  ? 

If  2  gallons  of  molasses  cost  8  dimes,  what  will  i  gallon 
cost  ? 

If  i  bushel  of  corn  costs  8  dimes,  what  will  i  peck  cost? 

(i  bushel  contains  4  pecks.  If  4  pecks  cost  8  dimes,  i 
peck  will  cost  2  dimes.) 

John  has  a  5-cent  and  a  3-cent  piece.  If  he  buys  3  tops 
at  2  cents  each,  how  much  money  will  he  have  left  ? 

A  merchant  has  a  piece  of  cloth  8  feet  long  from  which 
he  cuts  off  2  yards.  How  much  remains  ? 

A  yard  =  3  feet,  2  yards  will  =  6  feet.  If  he  cuts  off  6 
feet  there  will  remain  2  feet. 

(In  this  way,  as  soon  as  possible,  parts  of  the  tables  of 
compound  numbers  should  be  introduced,  and  the  chil- 
dren will  thus  gradually  become  familiar  with  the  entire 
tables.  Make  examples  from  every-day  life  which  will 
be  suggested  by  the  errands  the  children  must  do  or  by 
the  employment  of  their  parents.  Bring  their  knowledge 
of  number  into  immediate,  practical  use.) 


NINTH  STEP 


47 


NINTH  STEP. 

THE  NINE. 


The  Pure  Number. — Measuring. 


i+i+i+i+i+i+i+i+i=9 
9x1=9 

9—i—i—i—i—i—i—i—i=i 
9-*-  i  =9 


2+2+2+2+1=9 

4X2+1=9 

9  —  2  —  2  —  2  —  2  =  1 

9H-  2=4(1) 


48  THE  FIRST  YEAR:  i-io. 

:::j{  Xi-*=' 

«°-M  s^s-r1 
::::  if  «i±|=f 

1      9-4-4=1 
[     9  -i-  4  =  2  (i) 

(Measure  9  with  all  the  other  numbers  according  to  the 
plan  followed  with  the  preceding  numbers.) 

9  =  8+  i,  7  +  2,  6  +  3,  etc. 
8  =  9  —  i,  7  +  i,  6  +  2,  etc. 
7  =  9  —  2,  6  +  i,  5  +  2,  etc. 
6  =  9  —  3,  5  +  i,  4  +  2,  etc. 

(Continue  in  the  same  way ;  also  construct  tables  as  on 

P-  39-) 

9  =  9  x  i,  3  x  3. 


9  can  be  separated  into — 

3  equal  numbers,  3  +  3  +  3 ; 

4  equal  and  i  unequal  numbers,  2  +  2  +  2  +  2+1; 

2  equal  and  i  unequal  numbers,  4  +  4  +  i ; 

3  unequal  numbers,  2  +  3  +  4 ; 
2  unequal  numbers,  5+4. 

Rapid  Work. 

3X3— 3— 2x2— 5— 1x4? 
i +2+3-3—2+3—4? 
9-j-  3  +  4  +  2  —  1  -r-4  x  3? 

(These  should  be  given  very  rapidly,  and  should  involve 
all  kinds  of  operations  which  the  pupils  have  already  had. 
The  pupils  should  be  able  to  follow  as  rapidly  as  the 
teacher  can  dictate  the  combinations,  and  be  ready  with 
the  answer  as  soon  as  the  teacher  ceases.) 


NINTH  STEP.  49 


Combining. 

How  many  times  i  is  4  x  2  less  than  3x3? 

What  number  can  I  take  four  times  away  from  9  and 
have  i  left? 

What  part  of  the  6  is  the  third  part  of  9  ? 

Separate  9  into  two  unlike  numbers,  one  of  which  is 
i  greater  than  the  other. 

II.— The  Applied  Number. 

How  many  gallons  in  9  quarts  ? 

How  many  weeks  in  9  days  ? 

How  many  2's,  3's,  4's,  and  6's  in  9  cents  ? 

Mary  had  9  verses  to  learn.  She  learned  3  verses  each 
day.  How  many  days  did  it  take  her  ? 

Her  brother  wrote  9  pages  in  3  days.  How  many  each 
day? 

What  cost  3  sheets  of  paper  if  i  sheet  cost  3  cents  ? 

William  was  to  fetch  his  father  4  sheets  of  paper,  each 
sheet  costing  2  cents.  He  had  6  cents  and  3  cents.  How 
much  money  must  he  bring  back  ? 

The  milkman  asks  3  cents  a  pint  for  milk.  How  many 
pints  can  I  get  for  9  cents  ? 

I  give  Fannie  4  2-cent  pieces  and  a  i-cent  piece.  She 
gave  her  sister  3  cents  and  her  brother  4  cents.  How  much 
had  she  left  ? 

A  boy  buys  2  lemons  at  2  cents  each,  and  1  orange 
for  4  cents.  He  has  i  cent  left.  How  much  had  he  at 
first? 


SO  THE  FIRST  YEAR:  i-io. 

TENTH   STEP. 

THE    TEN. 


(We  have  now  reached  the  first  number  which  must  be 
considered  as  another  kind  of  unity,  or  another  kind  of 
One,— the  Ten.  So  we  write  again  the  figure  i,  but  to 
show  that  this  i  contains  ten  times  as  much  as  the  first  i, 
we  move  it  one  place  to  the  left,  and  say,  this  i  is  a  ten. 
The  vacant  place  of  the  simple  i  will  be  indicated  with  the 
cipher,  so — 10.  The  pupils  should  be  taught  as  follows  :) 

Show  me  10  fingers.  Now  i  finger.  Indicate  the  i 
finger  with  a  figure.  Indicate  the  fingers  of  both  hands 
with  a  figure. 

(The  children  are  shown  how  they  must  write  i  in  the 
ten's  place,  and  the  cipher  at  the  right  for  unit's  place, 
place.) 

(Ten  splints  also  may  be  used,  and  then  bound  into  a 
bundle  to  represent  i  ten.  This  must  be  continued'until 
the  pupils  comprehend  the  10  and  its  relation  to  the  unit.) 


TENTH  STEP.  5 1 

Measuring* 

(Grube  now  abandons  the  writing  out  of  the  tables, 
such  as  10  with  i,  10  with  2,  10  with  3,  etc.,  as  practised 
with  all  preceding  numbers.  The  teacher  can  require  the 
pupils  to  do  so  if  he  deems  it  desirable ;  but  it  will  prob- 
ably be  found  unnecessary  to  go  farther  in  this  direction.) 

The  10  consists  of  two  equal  numbers,  5  +  5 ;  of  5  equal 
numbers,  2  +  2  +  2  +  2  +  2;  of  2  equal  and  i  unequal 
numbers,  3  x  3  +  i  ;  of  4  unequal  numbers,  1+2  +  3  +  4. 

1  is  the  half  of  2  6  is  3  times  2 

the  third  of  3  2  times  3 

the  fourth  of  4,  etc.          9  is  9  times  i 

2  is  the  half  of  4  3  times  3 

the  third  of  6  8  is  8  times  i 

the  fourth  of  8,  etc.  4  times  2 

10  is  10  times  i  2  times  4 

5  times  2  7  is  7  times 

2  times  5  5  is  5  times 

3  is  the  half  of  6  4  is  4  times 

the  third  of  9  2  times 

4  is  the  half  of  8  3  is  3  times 

5  is  the  half  of  10  2  is  2  times 

6  is  6  times  i  i  is  i  times 

What  numbers  go  without  remainder  into  10,  9,  8,  6,  4? 
What  are  only  divided  by  i  and  themselves  ?      (The 
prime  numbers,  i,  3,  5,  7.) 

Rapid  Work. 

2x3  +  2  +  1—6  +  5  —  3x2-7-5? 

10  — 7x3  +  1-^-5x4+14-3  +  6? 

2x2  +  2  +  3  —  7x5-5-2  +  4-4-3? 

10— 2—  I— 2—  I—  2—  I? 

i  +  3  +  3  +  4? 

Combining. 

What  number  has  i  more  than  the  double  of  3  ? 
How  much  is  2  x  5  greater  than  the  difference  between 
3x3  and  the  double  of  4? 


52  THE  FIRST  YEAR:   i-io. 

A  father  divided  10  apples  among  his  4  children  so  that 
each  older  received  i  more  than  the  next  younger.  How 
many  did  each  receive  ? 

(The  10  consists  of  the  4  unequal  numbers,  i  +  2  +  3  +  4; 
each  is  i  larger  than  the  next  below.  Therefore  the 
father  could  give  the  youngest  i  apple,  the  next  2,  etc.) 

N.  had  learned  4  proverbs.  His  brother  said  to  him, 
"  I  know  twice  as  many  again  as  you  and  2  more."  How 
many  did  he  know? 

Herman  said,  "I  am  5  times  as  old  as  my  brother." 
The  brother  was  2  years  old.  How  old  was  Herman  ? 

II. — The  Applied  Number. 

In  10  pints  how  many  quarts  ? 

In  10  days  how  many  weeks  and  days  ? 

In  10  cents  how  many  2-cent  pieces?  3-cent  pieces? 
5-cent  pieces  ? 

Fred  had  6  cents,  3  cents,  and  i  cent.  He  went  to  a 
stationer  and  bought  4  sheets  of  paper  at  2  cents  a  sheet, 
and  2  sheets  at  i  cent  a  sheet.  Did  he  have  money 
enough  ? 

Karl  had  the  same  amount  of  money,  and  bought  3 
sheets  at  3  cents  each.  How  much  did  he  have  left  ? 

How  many  pints  of  milk  can  be  bought  for  10  cents  if 
i  pint  costs  2  cents?  5  cents? 

How  many  biscuits  can  I  buy  for  10  cents  at  2  cents 
each  ? 

A  dime  has  10  cents.  How  many  2-cent  pieces  equal  a 
dime  ?  How  many  5-cent  pieces  ? 

Ten  dimes  make  a  dollar.  How  many  2-dime  pieces  in 
a  dollar  ?  5-dime  pieces  ? 

This  completes  the  first  school  year,  and  the  most  im- 
portant steps  in  number  have  been  mastered.  One  year 
is  not  too  long  if  the  work  has  been  thoroughly  done.  Of 
course  the  child  knows  only  the  numbers  i  to  10 ;  but  he 
knows  them  and  can  use  his  knowledge ;  therefore  they 
are  of  some  value  to  him.  Of  what  use  to  the  child  if  he 


TENTH  STEP.  53 

count  to  100,  but  could  not  separate  the  number  9  into  its 
elements  and  use  them  ?  The  process  of  "  measuring  " 
must  be  thoroughly  mastered  by  the  child.  He  must  be- 
come so  thoroughly  acquainted  with  all  of  the  operations 
of  each  step  that  without  hesitation  he  can  perform  them. 
The  eye  becomes  trained  by  use  of  the  blocks  to  habits  of 
accuracy,  and  the  child  learns  to  be  attentive.  By  means 
of  the  eye  and  by  handling,  he  gains  an  idea  of  the  num- 
ber and  its  combinations.  Thus  the  objects  appeal  to  his 
senses,  and  he  is  soon  able  to  pass  over  from  the  number 
obtained  from  concrete  things  to  the  abstract.  When 
objects  no  longer  are  necessary  to  give  the  idea,  the  concrete 
objects  must  not  be  named,  as  that  withdraws  the  attention 
from  the  abstract  number  itself.  Having  mastered  the 
abstract  number,  the  child  is  able  to  apply  it  with  concrete 
examples.  This  will  not  be  difficult,  as  little  exam  pies  are 
made  from  the  every-day  relations  of  life. 

At  the  10,  if  not  before,  the  use  of  objects  should  be  aban- 
doned entirely.  The  child  is  now  able  to  gain  the  abstract 
idea  without  the  help  of  objects.  Objects  become  a  cum- 
brance  as  soon  as  the  child  can  do  without  them.  Some 
think  they  can  be  abandoned  after  the  4  or  5.  Each 
teacher  must  settle  the  question  with  the  class  he  may 
have.  When  the  class  are  able  to  get  the  idea  without  the 
objects,  then  is  the  time  to  give  them  up.  Certainly  that 
end  will  be  reached  when  the  10  is  completed. 

Reviews  must  be  frequent,  and  every  step  must  be  mas- 
tered before  proceeding  to  the  next.  After  knowledge 
has  been  obtained  by  illustration  and  observation  it  must 
be  thoroughly  memorized. 


54  THE  SECOND  YEAR:  10-100. 

THE  SECOND  YEAR:  10-1.00. 

(See  Appendix,  p.  90.) 


OBSERVATIONS. 

1.  Grube  says:  "  Fingers  and  lines  continue  to  be  used 
for  illustration.     One  can  well  say  that  Nature  has  given 
to  man  the  decimal  system  of  number  in  the  hand."* 

2.  The  procedure  in  the  following  steps  is  the  same  as 
that  given  for  the  smaller  numbers.     Multiplication  and 
division  should  be  given  both  as  written  and  oral  work, 
while  addition  and  subtraction  need  only  be  oral.     The 
pupils  must  continue  the  "  measuring"  of  each  new  num- 
ber by  the  numbers  from  i  to  10  until  the  greatest  mechan- 
ical skill  is  reached.     This  mechanical  skill  is  connected 
with  the  greatest  self-activity  on  the  part  of  the  pupil. 

3.  For  the  operations  with  the  pure  as  well  as  the  ap- 
plied numbers,  a  greater  diversity  in  the  manner  of  expres- 
sion in  the  examples  can  be  employed,  in  order  that  the 
pupil  may  become  more  and  more  free  from  the  formulas 
of  the  earlier  work.   Applied  examples  should  be  gathered 
from  the  pupil's  surroundings,  from  material  with  which 
he  is  familiar.     Here  is  an  excellent  opportunity  to  lead 
the  pupil  to  invent  examples,  and  the  privilege  of  giving 
an  example  to  the  class  may  be  accorded  to  the  pupil  first 
solving  a  given  example.     This  originating  of  examples 
will  not  be  difficult,  because  the  pupil  always  proceeds 
from  the  preceding  step,  and  only  adds  to  the  already 
known. 

*  While  Grube  continued  to  use  objects  to  illustrate  the  number, 
his  modern  followers  abandon  them,  as  we  have  already  shown.  Sim- 
ple lines  may  be  used  profitably,  as  the  following  pages  illustrate,  to 
show  the  relations  of  units  to  tens,  etc.  But  the  numbers  do  not  need 
to  be  longer  taught  by  the  use  of  objects,  as  heretofore. 


ELEVENTH  STEP.  $5 

ELEVENTH  STEP. 

THE    ELEVEN. 


I.— The  Pure  Number. 

10  times  one  or  10  ones  make  i  ten. 
If  I  have  10  ones  taken  together,  I  have  i  ten  and  no  (o) 
ones  or  units  besides. 

llllllllll  =  i  ten  and  o  ones  =  10. 

If  another  one  is  added,  it  belongs  to  the  second  ten. 

I  I  I  I  I  I  I  I  I  I  10+  i  =  11. 

What  is  the  i  at  the  right  ?  What  the  i  at  the  left  ? 
Where  does  the  one  (unit)  belong  ?  How  many  ones  must 
be  added  in  order  to  make  the  second  ten  full  ?  What  do 
we  call  i  ten  and  i  one  in  one  word  ?  What  is  the  1 1  ? 


50  THE   SECOND   YEAR:   10-100. 

Oral. 
Measure  with  i. 

i  +  i  +  i  +  etc.  =11.  (i  +  i  =  2, 2  +  i  =  3,  3  +  i  =  4, 
etc.) 

II   X   I  =  II. 

1 1  —  i  —  i  —  i,  etc.  =  i.  (i  i  —  i  =  10,  10  —  i  =  9,  9  T- 
i  =  8,  etc.) 
ii  -j-  i  =  ii. 

Measure  with  2. 

2  +  2  +  2  +  2+2+1=11. 

5X2  +  1  =  11. 

11—2  —  2  —  2—2  —  2  =  1. 


Measure  with  10. 

10  +  i  =  ii. 

i  x  10  +  i  =  ii  (i  ten  +  i  one  =  n). 

11  —  10  =  i. 

ii  -5-  10  =  i  (i)  (In  ii  is  i  ten  +  i  one). 

(Each  pupil  gets  by  this  means  a  principle,  and  as  he 
knows  the  course  to  follow,  all  assistance  from  the  teacher 
must  cease.) 

All  numbers  hereafter  are  measured  only  by  the  num- 
bers from  i  to  10. 


Written. 

\\  \\  \\  \\  11  =  10+  i 

=  ii 

1 

II  =  II    X    I                                    II 

-*-  i  =  ii 

5x2  +  1 

'  2  =  5(1) 

3x3  +  2 

3  =  3  (2) 

2x4  +  3 

4  -  2  (3) 

2x5  +  1 

5  =  2  (i) 

x  6  +  5 

6  =  i  (5) 

x  7  +  4 

7  =  i  (4) 

x  8  +  3 

8  -  i  (3) 

x  9  +  2 

9  =  i  (2) 

X  10+  I 

10  =  I  (I) 

ELEVENTH  STEP.  57 

Comparison  (oral). 
1 1  =  10  +  i,  9  +  2,  8  +  3,  etc. 
1 1  =  1 1  x  i,  i  =  ^  x  1 1  (i  is  ^  of  1 1). 
Form  1 1  from  3  equal  and  i  unequal  number, 

4  equal  and  2  unequal  numbers. 

5  equal  and  i  unequal  number. 
4  unequal  numbers. 

Rapid  Work. 

I  have  6  cents,  3  cents,  i  cent,  and  i  cent,  and  give  away 
4  cents,  2  cents,  and  3  cents.     How  much  have  I  left  ? 

ii  —  2  —  3  — 4  +  3  —  i-*-2  x  5? 

II— 2  —  I— 2  —  I— 2—  I? 
2X  5  +  1—9x4  +  3  —  7+4? 

ii  —  5  +  3  +  %  +  5  X4  — 3x2  +  1? 
(Let  all  the  possible  combinations  within  the  11  be  given 
orally  until  the  pupil  can  reckon  as  rapidly  as  the  teacher 
can  give  them.) 

Combining. 

How  many  I's  must  I  add  to  5  x  2  to  get  11  ? 

From  what  number  must  I  take  3  x  3  to  get  2  ? 

How  often  can  I  take  the  fourth  part  of  8  away  from  n? 

What  number  is  i  ten  greater  than  i  ? 

What  is  the  difference  between  4x2  and  1 1  ?  « 

II.  The  Applied  Number. 

n  cents  =  3  three-cent  pieces  and  i  two-cent  piece. 

I 1  pints  contain  5  quarts  and  i  pint. 
1 1  days  =  i  week  and  4  days. 

N.  made  a  journey  of  11  days  and  used  just  II  dollars. 
How  much  was  that  per  day  ? 

B.  used  on  a  journey  1 1  dollars.  If  he  used  i  dollar  a 
day,  how  many  days  was  he  on  the  way? 

Fanny  had  2  five-cent  pieces  and  i  cent.  She  bought  2 
lead  pencils  at  3  cents  each  and  2  at  2  cents  each.  How 
much  money  had  she  left? 

A  boy  was  given  2x4  +  3  cents  for  doing  errands.  He 
put  5  cents  +  3  cents  in  his  bank,  spent  2  cents  for  candy 


58  THE  SECOND   YEAR:    10-100. 

and  gave  the  rest  to  his  sister.  How  much  did  he  give 
her? 

If  milk  costs  4  cents  a  quart,  how  many  quarts  can  I  buy 
for  ii  cents  and  how  many  cents  would  I  have  left? 

A  mother  gave  one  son  3x2  cents,  and  another  2x2 
4-  i  cent.  How  much  did  she  give  both  ? 

Henry  had  5  apples,  John  gave  him  2  and  William  gave 
him  2x2.  How  many  had  he  then  ? 

Mr  A.  had  n  nuts  which  he  divided  among  four  boys. 
To  the  first  he  gave  two  and  to  the  others  each  an  equal 
number.  How  many  did  each  of  the  others  receive  ? 


TWELFTH  STEP. 

THE    TWELVE. 


I.  The  Pure  Number. 

I  I  I  I  I  I  I  II  1  =  10  +  2=12 

1 1 

Oral. 

i  +  i  +  i  +  i,  etc.,  =  12 

12  x  i  =  12 

12  —  i  —  i  —  i,  etc.,  =  i 

12-*-  I  =  12 


TWELFTH  STEP.  59 

2  +  24-2  +  2-1-24-2  =  12 

6X2=12 

12  —  2  —  2  —  2  —  2  —  2  =  2 
12  -T-  2  =  6 

3  + 3 +  3 +  3  =  12 

4  x  3=  12 

12-3-3-3  =  3 
12-5-3  =  4 

(Measure  in  the  same  manner  by  all  the  numbers  as  far 
as  10.) 

Written. 

12=  12   X    I  12  -5-  I  =  12 

6X2  2=6 

4x3  3=4 

3><4  4  =  3 

2X542  5  =  2  (2) 

2x6  6=2 

1x74-5  7  =  i  (5) 

1x84-4  8  =  i  (4) 

1x9  +  3  9  =  i  (3) 

I    X  10+  2  10  =  I  (2) 

(i  ten  +  2  units.)     (In  12  are  i  ten  and  2  units.) 

12  =  11  +  i,  10  +  2,  9  +  3,  etc. 

12  is  i  more  than  n,  2  more  than  10,  etc. 

1  is  the  twelfth  part  of  12. 

2  is  the  sixth  part  of  12. 

3  is  the  fourth  part  of  12. 

4  is  the  third  part  of  12. 
6  is  half  of  12. 

From  what  equal  numbers  can  12  be  formed?  From 
what  unequal  numbers  ? 

Form  12  from  3  numbers,  the  first  of  which  is  2  and  the 
following  always  increasing  by  2. 


60  THE  SECOND   YEAR:  10-100. 

Rapid  Work* 

2x2  +  2x2-5-4—1? 
2+3+3+2+2—4+4+4x2? 

From  12  apples,  one  half  are  eaten,  then  half  the  re- 
mainder, then  i.  How  many  remained? 

From  12  cents  take  away  i  three-cent  and  i  two-cent 
piece,  then  again  i  three-cent  and  i  two-cent  piece.  How 
much  remains. 

12  — 6  +  3  —  5  +  7  —  1-5-2  — 3? 
IO  +  2-T-4X2  +  3-5-3  +  9? 

8  +  2  +  2-5-3-7-2x6  —  11  +  7? 

5x2  +  2-^-6  +  7-5-3x4  —  7? 

6-5x12  — 9x3  —  i-v-4? 

Combining. 

The  third  part  of  12  is  what  part  of  8  ?  The  half  of  12 
is  how  many  times  3  ? 

What  is  the  difference  between  \  of  12  and  £  of  10? 
12  is  3  times  what  number? 
What  number  must  I  subtract  from  12  to  get  9? 

(As  9  +  3  =  12,  3  must  be  taken  from  12  to  get  9.) 
What  number  subtracted  from  12  leaves  4? 

n.  The  Applied  Number. 

12  things  make  a  dozen. 

12  months  make  a  year. 

What  part  of  a  dozen  are  6  things  ? 

What  part  of  a  year  are  6  months  ? 

3  months  =  a  quarter  of  a  year. 

4  months  =  a  third  of  a  year, 
How  many  gallons  in  12  quarts? 
How  many  quarts  in  12  pints? 

How  many  sixes,  fours,  threes,  twos  in  12? 

*  It  must  not  be  forgotten  that  the  exercises  of  "  Rapid  Work"  are 
to  be  worked  as  fast  as  the  teacher  dictates,  and  that  it  is  oral  work 
only.  The  object  is  to  gain  facility  in  head-work.  All  idea  of  paren- 
thetical expressions  is  excluded  here. 


TWELFTH  STEP.  6 1 

In  a  month  there  are  four  weeks.  If  a  boy  earns  12 
dollars  a  month,  how  much  does  he  earn  per  week  ? 

A  father  pays  2  dollars  a  month  for  his  son's  lessons. 
How  much  is  that  for  3  months?  How  much  for  6 
months  ? 

Charles  divided  12  cents  equally  among  4  poor  boys. 
How  much  did  each  receive  ? 

How  many  sheets  of  paper  at  3  cents  each  can  you  buy 
for  12  cents? 

(As  many  times  as  I  have  3  cents  I  get  a  sheet  of  paper. 
12  cents  =4x3  cents,  so  I  get  4x1  sheet,  or  4  sheets.) 

Illustrated. 

Ill o 

III o 

III o 

III o 

A  foot  contains  12  inches.  How  many  inches  in  •£  of 
afoot?  t?  $? 

What  part  of  a  foot  is  4  inches  ?    6  inches  ? 

A  troy  pound  contains  12  ounces.  What  part  of  a 
pound  is  3  ounces  ?  4  ounces  ? 

If  a  top  costs  3  cents,  a  whistle  2  cents,  and  a  ball  7 
cents,  how  much  do  all  cost  ? 

John  takes  a  dime  and  a  2-cent  piece  to  the  store  and 
buys  4  lead  pencils  at  2  cents  each,  and  a  sponge  for  3  cents. 
How  much  money  had  he  left  ? 

John  has  8  cents ;  how  much  more  must  he  earn  to  have 
a  dozen  cents  ? 


62  TtiE   SECOND   y&A&:  16-106. 


THIRTEENTH  STEP. 

THE    THIRTEEN. 


I  Illlll  I  I  1=  10  +  3=  13 
III 

(Measuring  and  rapid  work  the  same  as  before.) 

Combining. 
Make  13  by  multiplying  3*3  and  2's. 

(3x3  +  2x2  =  9  +  4=  13.) 

How  does  the  difference  between  13  and  9  compare  with 
the  difference  between  12  and  8? 
Subtract  6  from  13. 

(13  —  6  =  7,  for  13  —  3  =  10  and  10  —  3  =  7.) 
What  number  =  7  +  6  ? 
What  number  =  8  +  5  ? 
What  number  must  I  add  to  4  x  3  to  get  13  ? 

n.  The  Applied  Number. 

Mary  has  a  ro-cent  and  a  3-cent  piece,  with  which  she 
buys  4  oranges  at  3  cents  each.  How  much  has  she  left  ? 

A  gentleman  divides  13  apples  among  some  children, 
giving  the  first  child  3  apples,  and  the  others  2  apples 
each.  How  many  children  were  there  ? 


FOURTEENTH  STEP. 


FOURTEENTH  STEP. 

THE    FOURTEEN. 


I  I  I  I  I  I  I  I  I  I  =  10  +  4  =  14. 
I  I  I  I 

What  number  must  I  double  to  get  7x2? 

2  sparrows  lit  upon  a  tree,  and  then  2  more,  and  3,  and 
3  and  2x2;  3  +  4  +  5  soon  flew  away.  How  many  re- 
mained ? 

I  have  taken  a  certain  number  3  times  away  from  14, 
and  have  2  left.  What  is  the  number  ? 

(If  I  have  2  left,  I  must  have  taken  12  away ;  12  is  3  X4, 
so  I  have  taken  4  three  times  away  from  14  if  I  have  2 
remainder.) 

H.— The  Applied  Number. 

14  days  =  2  weeks. 

14  cents  =  7  two-cent  pieces,  4  three-cent  and  I  two- 
cent  piece,  etc. 

14  things  =  i  dozen  and  2  things. 

14  months  =  i  year  and  2  months. 

If  I  travel  for  2  weeks,  and  spend  i  dollar  each  day,  how 
much  will  my  journey  cost  ? 


64  THE  SECOND   YEAR:  10-100. 

What  will  i  dozen  and  2  pencils  cost  at  i  cent  each  ? 

How  many  sheets  of  paper  can  I  buy  for  i  dime  and  4 
cents,  if  a  sheet  costs  2  cents  ? 

Marie  knit  in  i  year  7  pair  of  stockings.  How  many 
dozen  did  she  knit,  and  how  many  stockings  over? 

If  I  sell  7  pigs  at  2  dollars  each,  how  much  do  I  receive  ? 

A  man  buys  14  pounds  of  sugar  for  i  dollar  and  4  dimes. 
What  does  i  pound  cost  ? 


FIFTEENTH  STEP. 

THE    FIFTEEN. 
I.— The  Pure  Number. 

II  I  I  I  I  I  I  I  I    10  +  5  =  15. 
I  I  I  I  I 

How  many  units  belong  to  the  second  10?  How  many 
are  still  lacking  to  make  the  second  10  full  ?  Write  15 
in  dots  so  that  5  always  stand  together. 


Write  these  fives  under  each  other. 
•    •    •    •    •    j 
f   3  x  5 

Write  them  so  that  3  stand  in  a  row. 


:f5X 


3 


Of  what  unequal  numbers  does  1 5  consist  ?    Of  what  3 
unequal  numbers  ? 
What  is  the  difference  between  15  and  8? 


FIFTEENTH  STEP.  65 

(a.  I  observe  how  much  I  must  add  to  8  to  get  15.  8 
-f  2  =  10,  10  +  5  =  15,  8  +  7  =  15.  The  difference  be- 
tween 8  and  15  is  7. 

b.  I   take  8  from  15;  15  —  5  =  10,   10  —  3  =  7,  15—8 

=  70* 

In  what  number  is  found  an  entire  and  a  half  ten  ? 

What  5  numbers  give  the  sum  of  15  ? 

Each  following  of  these  5  numbers  is  to  be  i  larger  than 
the  preceding.  1+2  +  3  +  4  +  5. 

At  Easter  a  mother  divided  among  her  5  children 
boiled  eggs  according  to  age,  so  that  each  older  got  i  egg 
more  than  the  next  younger.  The  middle  child  in  age 
got.  3  eggs.  How  many  did  each  of  the  others  get? 
How  many  eggs  were  distributed  ? 

II. — The  Applied  Number. 

Compare  i  doz.  with  15. 
i  doz.  =4x3  things. 

15  =  5  x  3  things  =  i  doz.  +  3  things. 
15  cents=  i  dime  +  5  cents. 
1 5  days  =  2  weeks  +  i  day. 

Mary  buys  15  pints  of  milk;  how  many  quarts  does 
she  buy  ? 

I  sent  a  friend  3  five-cent  postage  stamps.  How  much 
are  they  worth  ?  I  bought  4  two-cent,  2  one-cent,  and  i 
five-cent  stamp.  How  much  did  all  cost  ? 


*  A  strong  point  practised  by  the  Germans  in  their  arithmetical  cal- 
culations is  here  illustrated.  They  make  the  10  an  important  factor  in 
all  operations.  This  will  be  more  fully  discussed  later. 


66  THE   SECOND   YEAR:   10-100. 


SIXTEENTH  STEP. 

THE    SIXTEEN. 
I.— The  Pure  Number. 

I  I  I  I  I  I  I  I  1  I     10  +  6  =  16. 
Mill! 

How  many  ones,  twos,  threes  in  the  second  10  > 
Divide  the  16  lines  into  twos,  fours,  eights. 

I  I 
I  I 


Write  16  in  twos  perpendicularly. 

In  fours.  In  eights. 


4x4  2x8 

Where  do  we  find  equal  numbers  of  points  horizontally 
and  perpendicularly  ? 

(Many  pupils  will  now  be  able  to  construct  a  square  and 
divide  it  into  equal  parts.) 

Rapid  Work. 

How  many  are  2  +  2  +  2+2  +  2  +  2  +  2  +  2? 
How    many    are    1+2+3  +  1  +  2  +  3  +  1+2    less 
than  16? 

How  many  are  16  —  3  —  3  +  2  —3  ? 
16^4  +  3  +  2-5-3  x  5  —  9  +  10? 

Combining. 

How  do  you  find  the  half  of  16? 
(16  =  i  ten  +  6  units.    \  of  i  ten  is  5  units,  \  of  6  units 


SIXTEENTH  STEP.  67 

is  3  units.    5  units  +  3  units  =  8  units.    Therefore  i  of 
16  =  8.) 

A  had  6  dimes.    B  said,  "  If  you  take  4-  of  your  money 

8  times,  you  will  have  as  much  as  I."    How  much  had  B  ? 

(B's  amount  was  8  x  £  of  A's ;  i  x  6  dimes  [the  third 

part]  =  2  dimes.      Therefore  B  had  8x2  dimes  =  16 

dimes.) 

A.                                                   B. 
I  I  I  I  I  I  I  I  


What  number  =2  x  J  of  16  ? 
What  part  of  16  is  that? 

(J  of   16=  4,  2  x  4  =  8.    Therefore  8  is  2  x  J  of   16. 
16  =  2  x  8.    Therefore  8  =  J  of  16.) 


•I: 


II. — The  Applied  Number. 

(Applications  of  the  denominations  of  compound  num- 
bers should  be  made  as  soon  as  a  number  embraced  in  any 
table  is  reached.  If  possible,  place  the  various  measures 
in  the  hands  of  the  pupils  and  let  them  measure  out  the 
various  denominations.  Dots  may  be  used  to  give  a  pic- 
ture of  the  relations  of  the  various  denominations  to  each 
other.  In  this  way  all  the  tables  of  compound  numbers 
will  be  gradually  and  intelligently  learned.) 

In  i  bushel  there  are  4  pecks.  How  many  bushels  in 
1 6  pecks  ? 

A  pound  avoirdupois  contains  16  ounces.  How  many 
ounces  in  half  a  pound?  What  part  of  a  pound  is  4 
ounces  ? 

If  a  pail  holds  2  gallons,  how  many,  pailfuls  will  16  gal- 
lons make  ? 

A  farmer  sold  16  bushels  of  potatoes  at  half  a  dollar  a 
bushel.  How  much  does  he  get  for  all  ? 


68  THE  SECOND   YEAR:    10-100. 


SEVENTEENTH  STEP. 

THE  SEVENTEEN. 

I. — The  Pure  Number. 

MINIMI!     10  +  7  =  17. 
II  I  I  I  I  I 

How  many  ones  have  we  now  ?  How  many  are  lacking 
of  2  full  tens  ?  How  many  more  has  the  first  row  than 
the  second  ? 

With  what  numbers  can  we  measure  the  17?  Begin 
with  1 6. 

17  =  16  +  i 
15  +  2 
14  +  3,  etc. 

Of  how  many  ones,  twos,  etc.,  does  17  consist? 

17  =  17  x  i 

8x2  +  1 
5x3  +  2 
4x4  +  1,  etc. 

Make  17  from  3  equal  and  i  unequal  number. 

17=  3x5  +  2 
3X4  +  5 
3x3  +  8,  etc. 

Make  17  from  4  equal  and  unequal  numbers,  also  from 
5  equal  and  i  unequal  number. 

Rapid  Work. 

17  —  2—  2—  2  —  2  —  2  —  2  —  2? 

17-3-3-3-  etc. 
17  — 4  — 4  — 4,  etc. 
17-5-5.  etc. 


SEVENTEENTH  STEP.  69 

1  +  2  +  3+4+5  lacks  how  many  of  17  ? 

2  threes,  i  five,  and  i  three  lacks  how  many  of  17  ? 
17  —  7-^-5  x  8-7-4—1  x  5? 

3x5  +  2—1-5-4  +  8  +  5—1? 

Combining. 

How  many  ones  must  I  add  to  3  x  5  to  get  17  ? 

How  many  ones  must  I  add  to  5  x  3  to  get  17  ?  (An- 
swer: The  same.) 

What  relation  has  4x4  and  3  x  5  to  17  ? 

I  have  taken  4x4  from  17  and  obtained  just  the  same 
as  if  I  had  taken  double  another  number  from  17.  Of 
what  number  must  I  have  taken  the  double  ? 

II. — The  Applied  Number, 

How  many  pounds  in  17  ounces  ? 

Four  brothers  divided  17  cents  so  that  the  oldest  had 
i  one  more  than  the  others.  How  many  cents  did  he 
get? 

In  17  quarts  how  many  gallons  ? 

Charles  had  a  dime,  a  five-cent  piece,  and  a  cent.  How 
much  did  he  lack  of  17  cents  ? 

A  milkman  had  2  cans  of  milk,  each  holding  10  gallons. 
One  was  full,  and  the  other  had  7  gallons  in  it.  How  many 
gallons  had  he  in  both  ?  How  many  gallons  did  the 
second  can  lack  of  being  full  ? 

Henry  divided  17  cents  equally  among  5  poor  children. 
How  many  cents  did  each  get,  and  how  many  had  he 
left? 

If  Charles  can  walk  3  miles  in  an  hour,  how  long  will 
it  take  him  to  walk  17  less  2  miles? 


7O  THE   SECOND  YEAR:   10-100. 


EIGHTEENTH  STEP. 

THE    EIGHTEEN. 
I.— The  Pure  Number. 

I  I  I  I  I  I  I  I  I  I     10  +  8  =  18. 
1  I  I  I  I  I  I  I 

Write  the  number  18  in  dots  so  that  2  always  come  to- 
gether. 


How  many  pairs  has  the  first  ten  ?    How  many  pairs 
are  lacking  in  the  second  ten  ? 
Write  the  number  18  so  that  3  lines  come  together. 

Ill   III   III 
III   III   III 

How  many  threes  and  how  many  sixes  has  the  18  ? 
Write  the  sixes  in  horizontal  lines. 

I  I  I  I  I  I 

I  I  I  I  I  I 

I  I  I  I  I  I 
Write  the  18  in  fives. 


Of  what  2  equal  numbers  does  18  consist?    Of  what  3 
6,  9? 

Of  what  3  equal  together  with  i  unequal  number? 
Of  what  4  equal  together  with  i  unequal  number? 

Rapid  Work. 
Add  rapidly  2  twos,  i  three,  2  twos,  i  five  and  i  two. 

2  +  2  +  2  +  3  +  3  +  3  +3? 

Count  upwards  by  twos,  commencing  with  2  (2,  4,  6, 
etc.)  to  1 8. 

The  same  backwards. 
The  same  commencing  with  3.    Also  backwards, 


NINETEENTH  STEP.  7 1 

Combining. 

Of  what  number  is  18  sixfold  ? 

What  is  the  number  of  which  12  is  twofold  and  18 
threefold  ? 

What  part  of  12  is  this  number?     Of  18 ? 

What  part  of  1 2  is  1 8  greater  than  12? 

What  number  must  I  multiply  by  3  to  get  18  ? 

How  much  greater  is  the  double  of  9  than  the  clouble 
of  8,  7,6? 

II. — The  Applied  Number. 

If  a  pound  of  meat  costs  9  cents,  how  many  pounds  can 
be  had  for  18  cents  ? 

How  many  weeks  are  there  in  18  days  ? 

If  Mr.  A  works  3  weeks  at  i  dollar  a  day,  how  much 
does  he  earn  ? 

Fred  was  sent  to  market  with  18  dimes.  He  bought  4 
pounds  of  veal  at  i  dime  a  pound  ;  6  pounds  of  beefsteak 
at  2  dimes  a  pound ;  cabbage  for  i  dime.  How  much 
money  had  he  left  ? 

If  a  child  is  1 8  months  old,  how  many  years  old  is  it? 

A  farmer  had  18  pecks  of  clover  seed.  How  many 
bushels  had  he? 


NINETEENTH  STEP. 

THE  NINETEEN. 

I.— The  Pure  Number. 

1. 

II  I  I  I  I  I  I  I  I     10  +  9  =  19. 
I  I  I  I  I  I  I  I  I 

What  have  we  now  ? 

We  have  i  ten  and  9. 

How  many  does  this  lack  of  2  full  tens? 

Jt  lacks  but  i  and  then  the  second  ten  is  complete. 


2  THE   SECOND   YEAR:   10-100. 

Write  19  in  lines  of  2  each. 

II       II       11       II       II 
II       II       II       II       I 
Write  the  ten  in  twos  and  the  nine  in  threes. 


Write  in  lines  of  5  each. 


I  I  I 
I  I  I 


I        I  I  I  I  I 
I         I  I  I  I 


How  many  fives  in  19?   sixes?  sevens?  eights?    Il- 
lustrate these  by  lines. 

I  I  I  I  I 
I  I  I  I  I 
I  I  I  II 

II  I  I  I  I  I 
I  I  I  I  I  I  I 
I  I  I  I  I 

etc. 

Rapid  Work. 


19—1  —  2  —2,  etc. 

(An  excellent  practice  is  found  in  starting  from  a  given 
number  and  counting  upward  as  far  as  19  and  backward 
by  twos,  threes,  fours,  etc.  For  example  :  by  twos  from 
3;  as,  3,  5,  7,  9,  u,  13,  15,  17,  19;  19,  17,  15,'  etc.:  or  by 
threes  from  4;  as,  4,  7,  10,  13,  16,  19;  19,  16,  13,  etc.  This 
must  be  done  with  greatest  rapidity  and  without  hesita- 
tion.) 

Combining. 

5  times  what  number  +  4  times  what  number  make  to- 
gether 19? 

3  times  a  number  +1  =  19.     What  is  the  number  ? 

6  times  a  number  +1  =  19.    What  is  the  number  ? 
How  can  I  divide  19  apples  among  6  children  so  that 

at  least  5  get  the  same  number  ?     How  many  would  the 
sixth  get  ? 


TWENTIETH  STEP.  73 

II. — The  Applied  Number. 

Gussie  had  2  dollars,  or  20  dimes.  George  had  i  dollar 
and  9  dimes,  or  19  dimes.  How  much  does  George  lack 
of  having  as  much  as  Gussie  ? 

"  My  little  brother,"  said  Anna,  "  is  \\  years  old ;" 
"  And  mine,"  said  Bertha,  "  is  just  i  month  older."  How 
many  months  old  was  the  latter?  How  much  over  i 
year? 

A  cloak  requires  3  yards  of  cloth,  each  yard  costing  6 
dollars.  Reckon  also  i  dollar  for  the  velvet  collar. 
What  will  the  cloak  cost  ? 


TWENTIETH  STEP. 

THE    TWENTY. 
I. 

I    I    I    I    I    I    I    I    I    I       10  +  10  =  20. 
I    i    I    I    i    I    I    I    I    I 

Now  how  many  tens  have  we  ? 

Show  me  20  fingers.  (Let  two  children  hold  up  both 
hands.) 

Here  are  twenty  sticks.  How  many  bundles  of  tens 
can  we  make  from  them  ?  We  bind  them  together  and 
have  what  number? 

Write  20  so  that  the  lines  fall  in  twos. 

II       II       II       II       II 
II       II       II       II       II 

How  many  fours  are  there  ? 

The  10  can  be  separated  in  two  equal  parts ;  how 
many  the  20  ? 

1 4  x  5=20 


74  THE   SECOND   YEAR:    10-100. 

How  many  points  in  one  of  these  horizontal  lines  ?  In 
one  of  the  perpendicular  lines  ? 

Now  write  4  dots  in  a  horizontal  line ;  how  many  rows 
are  there  ? 


\  5  x  4  =  20 

J 
Of  what  equal  numbers  does  20  consist? 

20  x  i,  10  x  2,  5  x  4,  4  x  5,  2  x  10. 

Of  what  number  is  20  the  double  ?    The  fourfold  ?    The 
fivefold?    The  tenfold? 
What  part  of  20  is  2,  4,  5,  10? 

Measure  with  i. 

1,  2,  3,  4,  5,  6,  (In  which  ten  are  we?    How  many  does 
it  lack  of  being  full?),  7,  8,  9,  10,  n,  12,  13,  14,  15. 

(How  many  have  we  in  the  second  ten  ?  How  many  in 
all?  How  many  ones  must  we  take  to  fill  the  second 
ten?) 

20  X   I  =  20. 

20,  19,  1 8,  17,  (Stop!  how  many  ones  must  we  still  take 
away  before  we  get  to  the  first  ten  ?)  etc. 

20  -*•  i  =  20. 

I  can  take  i  away  from  20  twenty  times,  or  i  is  con- 
tained in  20  twenty  times. 

Measure  with  2. 

2,  4,  6,  8,  10,  12,  (How  many  times  2  have  we  now?)  14 
16,  (How  many  now?)  18,  20. 

10  x  2  =  20. 
20,  1 8,  1 6,  14,  etc. 

?0  -T-  2  =  10, 


TWENTIETH  STEP.  75 

Measure  with  3. 

3,  6,  9,  12,  15,  1 8,  (How  many  must  we  still  add  to  get 
20?)  20. 

6  x  3  +  2  =  20. 

How  many  times  can  I  take  3  from  20  ? 

(As  often  as  I  take  3  away,  we  will  count  a  finger.) 

20,  17,  14,  n,  8,  5,  2. 

20  -5-  3  =  6  (2),  etc. 

20  =  20   X    I  20  -*-  I  =  20 

IO  X  2  2  =  IO 

6  x  3  (2)  3  =  6  (2) 

5  ><4  4  =  5 

4x5  5=4 

3  x  6  (2)  6  -  3  (2) 

2  x  7  (6)  7  =  2  (6) 

2  x  8  (4)  8  =  2  (4) 

2  X  9  (2)  9  =  2  (2) 

2  X   10  10  =2 

Rapid  Work. 
How  much  is   (2  x  2)  +  (2  x  2)  +  (2  x  2)  +  (2  x  2) 

+  (2X2)? 

How  much  is  (3  x  2)  -f  (3  x  2)  +  (3  x  2)  less  than  20  ? 
Subtract  4,  3,  2,   i  from  20  and  again  4,  3,  2,  i,  and 
how  many  remain  ? 
20  —  13  —  6? 
20—  ii  ?  9?  8?  6?  4? 
How  many  dozen  and  how  many  units  in  20? 

20  -7-2  +  5-^-5x3  +  1  X2  = 
20  —  4-7-4  +  6  X2  —  12  -*•  8  ? 

Combining. 

4  times  5  =  2  times  what  number  ? 

5  times  4  =  4  times  what  number  ? 

What  is  the  difference  between  4x5  and  5x4?     Be- 
tween 4x5  and  4x4? 


?  THE   SECOND   YEAR:   10-100. 

A  gardener  divided  20  apples  among  children  so  that 
he  gave  each  child  the  same  number  and  still  had  2  to 
save  for  Fred.  How  many  did  each  child  get  ? 

(If  Fred  received  2,  there  remained  18  to  divide.  18 
can  be  divided  into  3  equal  parts,  etc.) 

II.— The  Applied  Number. 

20  things  =  a  score. 

20  cents  =  2  dimes. 

How  many  weeks  in  20  days  ? 

How  many  pounds  in  20  ounces  ? 

Charles  had  20  cents  to  spend.  He  bought  3  apples  at 
2  cents  each,  a  ball  at  10  cents,  gave  his  sister  2  cents,  and 
kept  the  balance.  How  much  did  he  keep  ? 

James  is  20  years  old  and  his  brother  Chester  is  3  years 
younger  than  he,  while  his  cousin  John  is  6  years  younger 
than  Chester.  How  old  is  John  ? 

2  dimes  is  how  many  times  5  cents  ? 

20  cents  equal  how  many  2- cent  pieces  ? 

A  farmer  had  20  pecks  of  corn.  How  many  bushels 
had  he  ? 

A  board  is  20  inches  long.  How  much  of  it  must  I  saw 
off  in  order  to  leave  j  ust  a  foot  ? 

A  merchant  found  just  20  feet  in  a  piece  of  cloth.  How 
many  yards  did  the  piece  contain  ? 

A  milkman  had  2  ten-gallon  cans  full  of  milk.  He  sold 
ii  gallons  in  the  morning,  and  the  rest  in  the  evening. 
How  much  did  he  sell  in  the  evening? 

If  I  can  walk  4  miles  in  an  hour,  how  long  will  it 
take  me  to  walk  20  miles  ? 

How  long  if  I  walk  5  miles  an  hour  ? 

In  the  morning  there  were  9  boys  and  11  girls  at  school. 
In  the  afternoon  3  pupils  stayed  out.  How  many  were 
there  in  the  afternoon  ? 

If  I  pay  i  dollar  for  2  bushels  of  potatoes,  how  much 
must  I  pay  for  20  bushels? 

In  a  score  of  years  how  many  birthdays  will  you  have  ? 

(In  this  manner  all  the  foHowing  numbers  are  treated, 
and  the  teacher  will  now  be  prepared  to  continue  the 


THIRTIETH  STEP.  77 

course  himself.  A  written  preparation  should  be  made 
in  order  that  nothing  be  omitted,  and  that  the  pupil  be 
induced  in  the  best  manner  to  prepare  the  exercises  him- 
self. Especial  attention  should  be  given  to  such  numbers 
as  24,  30,  50,  60,  etc.,  which  are  more  often  applied. 
Such  numbers  as  23,  29,  etc.,  need  but  little  attention. 
Two  or  three  steps  more  will  suffice.) 


THIRTIETH  STEP. 

THE    THIRTY. 
I.— The  Pure  Number. 

I  I  I  I  I  I  I  I  I  I     10  -f  10  +  10  =  30. 

I  I  I  I  I  I  I  I  II 

I  I  I  I  I  I  I  I  I  I 

(3  times  the  fingers  of  2  hands.) 
If  I  add  i  to  the  29  the  third  ten  will  be  full. 
3  tens  taken  together  is  called  30. 

Measure  with  i. 

Count  by  ones  upwards:  i,  2,  3,  4,  5,  6,  7,  (Stop!  a 
pupil  says  :  "  We  are  in  the  first  ten  ;  it  lacks  3  of  being 
full,  and  20  more  of  30.")  8,  9,  10,  11,  12,  13,  (We  are  in 
the  second  ten,  etc.)  14,  15,  etc. 

30  x  i  =  30. 

Count  downwards  :  30,  29,  28,  (We  are  in  the  third  ten  ; 
from  the  third  ten  2  have  been  taken  away.)  27,  26,  etc. 

30  -j-  i  =  30. 

Measure  with  2. 
2,  4,  6,  8,  10,  12,  14,  (We  are  in  the  second  ten ;  it  lacks 


THE  SECOND  YEAR:   10-100. 


3  twos  of  being  full,  and  5  twos  more  of  completing  the 
3  tens  or  thirty.)  etc. 

15  x  2  =  30. 

Count  downwards  with  2 :  30,  28,  26,  24,  (We  are  in 
the  third  ten,  have  taken  3  twos  from  the  third  ten  and 
there  remain  2  twos  in  the  third  ten.) 

30 +-2=  15. 
Continue  with  the  other  numbers. 

Measure  with  10. 

10  +  10  =  20,  20  +  10  =  30. 

3  x  10  =  30. 
30  —  10  =  20,  20  —  10  =  10. 

30  -f-  10  =  3. 


3o  = 

30  x  i 

15x2 

10  x  3 

7  x  4  (2) 

6x5 

5x6 

4  x  7  (2) 

3  x  8  (6) 

3  x  9  (3) 

3  x  10 

Divide  the  dots  into  twos : 


Into  threes : 


3  x  10  =  30. 

30  -h  i  =  30 

2  =  15 

3  =  10 

4  =  7(2) 

I  =  6 

6  =  5 

7  =  4(2) 

8  =  3  (6) 

9  =  3  (3) 
10  =  3 


(•    30=  15   X  2. 

>•   30  =  10  x  3. 


THIRTIETH  STEP.  79 

Oral  : 

30  =  29  +  i,  28  +  2,  27  +  3,  etc. 

30  is  3o-fold  of  i  i  is  ^  of  30 

15-fold  of  2  2  is  TV  of  30 

lo-fold  of  3  3  is  TV  of  30 

6-fold  of  5  5  is  £  of  30 

5-fold  of  6  6  is  ^  of  30 

3-fold  of  10  10  is  £  of  30 

2-fold  of  1 5  1 5  is  i  of  30 

Of  what  equal  numbers  does  30  consist  ? 
Of  what  2,  3,  4,  5,  6  unequal  numbers  ? 

Rapid  Work. 

30-4-15  +3x5+  5-5-10? 

(3  x  5)  +  (2  x  4)  +  7  •*•  10  x  3  ? 

4x6,  the  half,  again  the  half,  x  5  ? 

Combining. 
30  -  19. 

(19  =   10  +  9,  30  —  10  =  20,  20  —  9  =  II,  30—  19=  II.) 

How  do  you  get  the  double  of  15  ? 

(15  =  i  ten  and  5  units ;  2  x  i  ten  =  2  tens  52x5 
units  =  10  units  =  i  ten  ;  2  tens  +  i  ten  =  3  tens  =  30.) 

Compare  30  with  16. 

(30  =  3  tens ;  16  =  i  ten  +  6  units.  I  must  add  to  the 
6  units  4  units  to  complete  the  second  ten,  and  still  i  ten 
to  get  3  tens.  Therefore  30  has  i  ten  +  4  units  =  14 
more  than  16,  and  16  is  i  ten  +  4  units,  or  14  less  than 
30.)* 

10  x  3  =  6  times  what  number? 

If  I  take  3x5  from  a  number,  I  get  5  x  3  as  remain- 
der. What  is  the  number  ? 

*  This  method  of  adding1  and  subtracting  is  universally  practised  at  pres- 
ent in  the  German  schools,  with  most  excellent  results.  The  tens  are  the 
stepping-stones  of  the  method. 


SO  THE   SECOND   YEAR:   10-100. 

II. — The  Applied  Number. 

In  30  days  how  many  weeks  ?  How  many  dozen  ?  How 
many  score  ? 

If  it  costs  me  2  dollars  a  day  when  I  travel,  how  many 
days  can  I  travel  and  spend  30  dollars  ? 

A  dollar  contains  10  dimes.  How  many  dollars  in  30 
dimes? 

A  workman  received  6  dollars  a  week.  How  much  will 
he  receive  in  5  weeks  ? 

A  piece  of  linen  was  10  yards  long.  How  many  feet 
long  was  it  ? 

Charles  had  a  cane  2  feet  6  inches  long.  How  many 
inches  long  was  it  ? 

William  had  7  gallons  and  2  quarts  of  water  in  a  tub. 
How  many  quarts  did  he  have  ? 

If  a  shirt  require  3  yards  of  cloth,  how  many  shirts  can 
be  made  from  a  piece  containing  30  yards  ? 

A  golden  eagle  equals  10  dollars.  If  Joseph  had  3 
eagles,  how  many  dollars  had  he  ? 

Mr.  A.  sold  3  sheep  at  5  dollars  each,  and  4  pigs  at  3 
dollars  each.  How  much  did  he  lack  of  getting  30  dol- 
lars? 

Five  francs  make  i  dollar.  How  many  dollars  are  equal 
to  30  francs  ? 

A  teacher  divided  30  apples  among  his  pupils,  giving  the 
boys  half  and  the  girls  half.  The  boys  received  each  3 
apples,  and  the  girls  each  5.  How  many  pupils  had  he  ? 


FIFTIETH  STEP. 

THE   FIFTY. 


X  10  =  50 


FIFTIETH  STEP.  8 1 

How  many  ones    stand    perpendicularly  under    each 
other  ? 

Write  these  fives  in  horizontal  lines. 
Write  50  in  twos.    How  many  stand  in  a  line  ? 
(Continue  as  in  preceding  numbers.) 

Oral  and  Written. 

50  =  50  x  i  50  -*-  i  =  50 

25  x  2  2  =  25 

16  x  3  (2)  3  =  16  (2) 

12x4(2)  4=12(2) 

10  x  5  5  =  10 

8  x  6  (2)  6  =  8    (2) 

7  x  7  (i)  7  =  7    (i) 

6  x  *8  (2)  8  =  6    (2) 

5  x  9  (5)  9  =  5    (5) 

5  x  10  10  =  5 

Rapid  Work. 

50  -i-  2  -f-  5  x  6  —  1 5  -*-  3  ? 
10  x  5  —  10  —  10  —  10  —  10  —5  ? 
25  x  2  -T-  5  + 15  —  5  +  20  +  10  ? 
30  -r-  2  x  3  +  5  -i-  5  +  40  -*-  50  ? 
50 -2-4-4-^-4  +  5x3- 15? 

Combining. 

The  nfth  part  of  50  is  double  what  number  ? 
The  half  of  50  is  5  times  what  number  ? 
How  does  \  of  50  compare  with  \  of  25  ? 
£  of  25  =  TV  of  what  number  ? 

II.— The  Applied  Number. 

How  many  weeks  in  50  days  ? 
How  many  pounds  avoirdupois  in  50  ounces  ? 
If  I  have  50  cents,  how  many  lo-cent  pieces  does  it 
equal  ?     How  many  5-cent  pieces  ? 
In  50  inches  how  many  feet  ? 


82  THE  SECOND  YEAR:   10-100. 

Charles  receives  from  his  aunt  a  2$-cent  piece;  from 
his  father  he  received  another.  How  many  cents  did  he 
get? 

A  milkman  shipped  to  New  York  5  cans  of  milk,  each 
containing  10  gallons.  How  many  gallons  did  he  ship? 

Mary  buys  50  cents'  worth  of  muslin,  paying  10  cents  a 
yard.  How  many  yards  does  she  get  ? 

A  coal-dealer  invested  50  dollars  in  coal  at  4  dollars  a 
ton.  How  many  tons  did  he  buy  ? 


HUNDREDTH  STEP. 
(The  Last  Step  of  the  First  Course.) 

THE    HUNDRED. 

The  counting  upwards  and  downwards  with  numbers 
from  i  to  10,  beginning  with  both  i  and  2,  must  be  done 
without  hesitation,  rapidly  and  accurately.  In  counting 
in  concert,  the  teacher  should  frequently  stop  the  pupils 
with  questions  as  before.  To  illustrate  : 

Counting  with  2,  beginning  with  i :  i,  3,  5,  7,  9,  11, 
13,  (Stop  !  In  what  ten  are  we?  How  many  units  does 
it  lack  of  being  complete  ?  How  many  tens  still  remain 
to  complete  100?  How  many  units  ?)  15,  17,  19,21,  etc. 

Counting  with  6,  beginning  with  2  :  2,  8,  14,  20,  26, 
32,  (Stop!  In  what  ten  are  we?  How  many  units  are 
lacking  to  complete  the  ten?  How  many  tens  to  com- 
plete ico  ?)  38,  44,  50,  etc. 

Counting  with  6,  beginning  with  99  :  99,  93,  87,  81, 
(Stop !  Ask  questions  as  above.) 

(Count  both  upwards  and  downwards  with  all  numbers 
from  i  to  10  in  this  manner.  This  will  be  found  a  most 
valuable  exercise.  If  the  drill  has  been  faithfully  kept  up 
with  all  preceding  numbers,  it  will  now  be  very  easy  and 
satisfactory.) 


HUNDREDTH  STEP. 

Written. 
Pupils  should  write  the  above  tables  as  follows : 

1+2  =  3  or      1+7=8  or      ioo  —  8  =  92 

3  +  2  =  5  8  +  7  =  15  92  —  8  =  84 

5  +  2  =  7  15+7  =  22  84  —  8  =  76 

7  +  2  =  9  22 +  7  =  29  etc. 

etc.    '  etc. 


10  x  10  =  100 


100  =  loo  x  i 
50  x  2 
33  x  3  (i) 
25  x  4 
20  x  5 
16  x  6  (4) 
14  x  7  (2) 
12  x  8  (4) 
ii  x  9  (i) 

IO  X  IO 


100  •*•  I  =  100 

2  =  50 

3  =  33  CO 

4  =  25 

5  =  20 

6  =  16  (4) 

7  =  14  (2) 

8  =  12  (4) 
9=11  (i) 

10  =  10 


ioo  =  99  +  i,  98  +  2,  97  +  3,  etc. 


100  is  loo-fold  of  i 
5o-fold  of  2 
25-fold  of  4 
2o-fold  of  5 
5-fold  of  20 
2-fold  of  50 


1  is  TJ-g-  of  ioo 

2  is  TV 

4  is  A- 

5  is  A 
20  is  \ 
50  is  i 


84 


THE   SECOND  YEAR:   10-100. 


Rapid  Work. 

IOO  -*-  2  +  IO  -4-  30  X   25  X  2  ? 

4  x  25  -  50  -*-  5  x  10  -  75  x  3  +  25  ? 

All  possible  combinations  in  multiplication  have  al- 
ready been  learned  in  the  course  of  the  various  exercises, 
but  may  now  be  arranged  in  the  multiplication  table  and 
practised. 


1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

2 

4 

6 

8 

10 

12 

H 

16 

18 

20 

3 

6 

9 

12 

15 

18 

21 

24 

27 

30 

4 

8 

12 

16 

20 

24 

28 

32 

36 

40 

5 

IO 

15 

20 

25 

30 

35 

40 

45 

50 

6 

12 

18 

24 

30 

36 

42 

48 

54 

60 

7 

14 

21 

28 

35 

42 

49 

56 

63 

70 

8 

16 

24 

32 

40 

48 

56 

64 

72 

80 

9 

18 

27 

36 

45 

54 

63 

72 

81 

90 

10 

20 

30 

40 

5o 

60 

70 

80 

90 

IOO 

HUNDREDTH  STEP. 


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8 

86  THE   SECOND   YEAR:   10-100. 

The  multiplication  table  can  also  be  written  in  rows  as 
follows : 

2  x  1=2;  2  X2  =  4;  2x3  =  6,  etc.; 
3Xi=3;3X2  =  6;3X3  =  9,  etc.; 
4  x  i=4;4X2  =  8;4X3  =  i2,  etc.; 

and  so  forth  until 

10  x  i  =  10;  10  x  2  =  20;  10  x  3  =  30,  etc. 

The  pupil  must  learn  that  the  product  is  the  same  when 
the  factors  are  alike,  no  matter  in  what  order  taken. 

7x6  =  42,  6x7  =  42. 

Besides  what  has  already  been  given,  the  four  funda- 
mental rules  must  be  practised  with  numbers  above  10 
until  the  pupils  are  brought  to  complete  mastery  of  them, 
and  are  able  to  make  various  combinations  with  great 
rapidity  and  accuracy.  A  few  examples  follow  which 
must  be  worked  only  orally  : 

a.  (Addition) — 

14  +  13  +  12  +  ii 

15  +  17  +  19  +  18 
25  +  37  +  39  +  17 
42  +  15  +  26  +  37 

etc. 

(The  method  of  working  is  as  follows :  42  +  10  =  52, 
S2  +  5  =  57,  57  +  20  =  77,  77  +  6  =  83,  83  +  30  =  113, 
113  +7  =  120.) 

b.  (Subtraction) — 

90—19—12  —  11 
98  -  32  -  41  -  24 
90  —  16  —  17  —  28  —  29 
97  —  12  —  34—  16  —  27 
etc. 

(To  be  worked  as  follows :  97  —  10=  87,  87  —  2  =  85, 
85-30  =  55,  55-4  =  5i,  51  -10  =  41,  41-6  =  35, 
35-20=  15,  15-7=8.) 


HUNDREDTH  STEP.  87 

c.  (Multiplication) — 

3  x  30,     4  x  22,     2  x  44,     2  x  27, 
3  x  25,    4  x  18,  12  x  5,     33  x  2, 

35  x  2,    45  x  3,     15  x  7,     19  x  4, 

42  x  6,     17  x  9, 

etc. 

(Employ  the  following  method  of  solution:  (17  x  9) 
10  x  9  =  90,  7  x  9  =  63  ;  90  +  60  =  150,  150  +  3  =  153. 
(Likewise  15  x  37)  10  x  37  =  370,  5  x  30  =  150,  5x7 
=  35  ;  37o  +  ioo  =  470,  470  +  5°  =  52o>  52o  +  35  =  555-) 

d.  (Division) — 

60-5-3,    69-5-3,    96-5-4,     72  -5-4, 
84  •*-  4,    84-5- 12,  68  -*- 13,  72  -*- 18, 
53  -1-4,    62  -i-  5,    72  -*-  3. 

(Method  as  follows :  72  -f-  3  is  60  -5-  3  =  20  and  12  -*-  3  = 
4.     Thus  72  [60  + 12]  -j-  3  =  24  [20  +  4]. 
Another :  84  -*-  12  is  60  -5- 12  =  5  and  24  •*- 12  =  2.  Thus 

84  [60  -4-  24]  -*-  12  =  7  [5+2]. 

Another :  40  -s-  18  is  36  -5- 18  =  2.  Thus  40  •*•  18  =  2  (4).) 

Combining. 

The  teacher  will  write  a  row  of  figures  on  the  board 
to  be  added,  subtracted,  multiplied,  and  divided,  placing 
the  signs  between.  He  then  points  to  the  exercises,  and 
the  pupils  solve  them  rapidly.  The  teacher  must  always 
solve  the  examples  himself.  This  holds  good  for  all 
work  in  arithmetic,  and  is  very  important. 

Such  examples  as  the  following  must  be  rapidly  worked 
by  the  pupils  : 

(3  x  29)  —  (4  x  1 6)  +  7  -s-  10  x  9  x  3  ? 
(15  x  6)  -(45  -r-9)  x  37? 
(4  x  20)  —  1 5  —  5  -*-  20  x  45  ? 

90  —  45  —  5    X   2  +  20  -T-  10  +  6  X   6? 

As  a  test  whether  every  number  is  comprehended  by 
the  pupil,  let  him  name  the  factors  of  numbers  from  i  to 
ioo  without  naming  the  numbers.  Prime  numbers  can 
only  be  measured  with  i.  For  example:  i  x  i,  2  x  i, 
3  x  i,  2  x  2,  5  x  i,  2  x  3, 7  x  i,  2  x  4,  or  2  x  2  x  2,  3  x  3, 
etc. 


88  THE   SECOND    YEAR:   10-100. 

It  is  important  that  the  children  know  readily  that 
52  =  4  x  13,  68  =  4  x  17,  95  =  5  x  19,  and  so  on  to  100. 

Such  exercises  as  the  following  should  be  thoroughly 
drilled : 

2X11  =  22  2X12  =  24  2X13  =  26 

3X11=33  3x12  =  36  3X13  =  39 

4  x  ii  =  44  4  x  12  =  48  4  x  13  ='52 

5  x  ii  =  55  5  x  12  =  60  5  x  13  =  65 

etc.  etc.  etc. 

2  x  14  =  28 

3  x  14  =  42,  and  so  on  till  5x19  =  95. 

The  prime  and  composite  numbers  can  be  arranged  as 
follows : 

1  prime  number 

2  " 

3  " 

4  =  2X2 

5  prime  number 

6  =  2x3 

7  prime  number 

8  =  2X2X2 

9  =  3X3 

10  =  2  x  5 

11  prime  number 

12  =  2x2x3 

13  prime  number 

14  =  2  x  7 

15  =  3  x  5 

16  =  2X2X2X2 

17  prime  number 

18  =  3  x  3  x  2 

and  so  forth  till  100.     This  must  be  thoroughly  drilled. 

Combining. 

A  man  gave  away  J  of  100  dollars,  and  then  \  of  the 
rest.  What  part  of  the  whole  amount  had  he  still  ? 

I  have  taken  a  number  3  -times  and  obtained  4  more 
than  \  of  loo.  What  was  the  number  ? 

5  times  what  number  is  5  less  than  100  ? 


HUNDREDTH  STEP.  89 

(The  number  which  is  5  less  than  100  is  95.  If  95  is  5 
times  a  number,  that  number  is  contained  5  times  in  95. 
£  of  95  =  19.) 

75  is  3  times  J  of  what  number  ? 

50  is  4  of  4  times  what  number  ? 

60  is  6  times  T^  of  what  number  ? 

II.— The  Applied  Number. 

100  cents  =  how  many  dimes  ?  quarters  ?  halves  ? 

loo  feet  =  how  many  yards  ? 

100  days  =  how  many  months  (of  30  days)? 

loo  months  =  how  many  years  ? 

loo  quarts  =  how  many  gallons  ? 

100  pints  =  how  many  quarts  ? 

loo  things  =  how  many  dozen  ?  score  ? 

loo  ounces^  how  many  pounds? 

100  days  =  how  many  weeks? 

Charles  received  4  quarters  as  a  birthday  present.  He 
spent  50  cents  for  fruit  and  nuts,  gave  20  cents  away,  and 
lost  10  cents.  How  much  had  he  left  ? 

How  many  days  in  100  hours  ? 

How  many  hours  in  100  minutes  ? 

A  farmer  had  100  pecks  of  clover  seed.  How  many 
bushels  had  he  ? 

Mary  had  a  cord  100  inches  long.  How  many  feet  is 
that  ? 

Mrs.  A.  took  8  dozen  eggs  to  the  store.  How  many  eggs 
did  she  take  ? 

My  grandfather  is  fourscore  and  four  years  old.  How 
many  years  is  that  ? 

January  has  31  days,  February  28,  and  March  31.  How 
many  days  do  they  lack  of  100  ? 

Mr.  Thomas  sold  4  cows  at  $25  each,  and  laid  out  the 
money  in  sheep  at  $5  a  head.  How  many  sheep  did  he 
buy? 

How  many  years  in  \  a  century?  \ ?  -j^-? 

If  a  train  of  cars  goes  20  miles  an  hour,  how  long  will  it 
take  to  go  100  miles  ? 

A  ship  sailed  2  leagues  an  hour.  How  long  did  it  take 
her  to  sail  96  miles  ? 


9°  THE   SECOND   YEAR:    10-100. 

If  I  saw  a  board  100  inches  long  in  4  equal  pieces,  how 
long  will  each  piece  be  ? 

(Examples  of  this  kind  should  be  made  by  both  teacher 
and  pupils  until  the  pupils  are  able  to  thoroughly  apply 
the  number  100.) 

(This  completes  the  work  of  the  first  two  years  accord- 
ing to  Grube.  Later  educators  in  Germany  accept  the 
first  year's  work,  but  modify  the  second  very  materially. 
We  append  herewith  a  method  which  meets  with  very 
general  favor  in  Germany.  The  two  methods  must  not 
be  mixed,  and  no  attempt  should  be  made  to  combine 
them.  Accept  one,  reject  the  other.) 


APPENDIX  70  THE  SECOND  YEAR'S  WORK, 

The  decimal  system  is  the  basis  of  all  arithmetical 
operations  after  the  10  (the  first  year's  work).  The  chil- 
dren should  be  introduced  to  this  system  the  second  year. 
They  must  not  longer  consider  each  number  as  an  indi- 
vidual, but  as  a  part  of  a  system.  Accordingly,  the  fol- 
lowing should  be  the  plan  of  procedure. 

SECTION    L 

^Development  of  the  numbers  from  10-100  in  the 
pure  tens,  and  practice  of  the  four  fundamental 
operations  within  this  range. 

FIRST    UNITY. 
(Addition  and  Subtraction.) 

First  Step. 

a.  i  +  2  =  3  10  —  2  =  8 

3+2=5  8-2=6 

5+2=7  6—2=4 

7+2=9  4-2=2 


APPENDIX.  QI 

1+3=4  10  —  3  =  7 

4  +  3=7  7-3  =  4 

7+3  =  10  4-3  =  1 

The  same  with  4  and  5. 

b.  We  wish  to  count  100  miles.     Children  count  10  on 
the  numeral  frame.    That  we  will  call  10  miles.     Again 
10  miles  must  be  counted. 

10  miles  +  10  miles  =  20  miles. 

We  must  count  very  much  farther  before  we  get  100 
miles.     Count  another  10  on  the  frame  : 

20  miles  +  10  miles  =  30  miles. 
Continue  in  this  way  until  we  get 

90  miles  +  10  miles  =  100  miles. 

In  the  same  manner  reverse  the  process,  commencing 
at  100  and  subtracting  10  each  time : 

10  m.  +  10  m.  =    20  m.  100  m.  —  lorn.  =  90  m. 

20  "    +  10  "    =    30  "  90  "  —  10  "    =  80  " 

30  "    +  10  "    =    40  "  80  "  —  10  "    =  70  " 

40  "    +  10  "    =    50  "  70  "  —  10  "    =  60  " 

till  till 

90  m.  +  10  m.  =  100  m.  20  m.  —  10  m.  =  10  m. 

c.  Run  rapidly  through  the  tens  forwards  and   back- 
wards, using  the  numeral  frame  or  fingers: 

10  100 

20  90 

30  80 

40  70 

till  till 

100  10 

d.  With  the  frame  develop  the  ordinals. 
(Children  say  as  the  teacher  indicates.) 

That  is  the  first       10  miles; 

"        second  10     " 

"        third     10     " 
till 

That  is  the  tenth     10  miles. 


92  THE   SECOND   YEAR:    10-100. 

Or, 

That  is  the  tenth     10  miles  ; 
"         ninth     10      " 
etc. 

e.  Writing  the  tens  forwards  and  backwards  from  dic- 
tation: 

10  100 

20  90 

30  80 

etc.  etc. 

Second  Step. 

a.  Oral  practice  in  intervals  of  20,  30,  40,  forwards  and 
backwards : 

20  +  20  =  40  loo  —  20  =  80 

40  +  20  =  60  80  —  20  =  60 

60  +  20  =  80  60  —  20  =  40 

80  +  20  =  100  40  —  20  =  20 

10  +  20  =  30  90  —  20  =  70 

3O  +  2O  ==  50  70  —  2O  =  50 

50  +  20  =  70  50  —  20  =  30 

70  +  20  =  90          30  —  20  =  10 

2O  +  30  =  50  80  —  30  =  5O 

50  +  30  =  §0  50  —  30  =  20 

10  -f  30  =  40  ioo  —  30  =  70 

40  -f  30  =  70  70  —  30  =  40 

70  +  30  =  ioo  40  —  30  =  10 

In  this  same  manner  with  the  other  numbers. 

b.  Practise  writing  from  dictation  after  each  group. 

c.  Exercises  like  the  following : 

1  +  2=3  10  —  2    =8 

10  +  20  =  30  IOO  —  20  =  80 

2  +  3    =5  7-4   =3 
20  +  30  =  50                     70  —  40  =  30 

4  +  3    =7  8-5    =3 

40  +  30  =  70  80  —  50  =  30 


APPENDIX.  93 

d.  Concrete  examples : 

A  man  travelled  30  miles,  and  afterwards  20  miles  more. 
How  far  did  he  go  ? 

John  had  70  miles  to  go.  After  he  had  gone  40  miles, 
how  much  farther  had  he  still  to  go  ? 

(Other  examples  of  this  kind.) 


Third  Step. 

The  pupils  have  now  learned  that  the  tens  progress  the 
same  as  the  units,  that  they  can  be  added  and  subtracted 
the  same  as  the  units,  and  they  have  also  learned  how  to 
write  the  tens. 

Fourth  Step. 

a.  10  miles  +  20  miles  +  30  miles  =  ? 
30    "      +40     "      +20     "     =  ? 
50    "      +30     "      +20     "      =  ? 

etc. 

b.  90  miles  —  30  miles  —  20  miles  =  ? 
loo    "     —  40     "     —  30     "     =  ? 

80    "      —  10     "     —  50     "      =  ? 
etc. 

C.      30  +  40  —  50  +  10  —  20  =  ? 
70  —  30  —  20  +  60  +  10  =  ? 

60  +  40  —  50  —  30  +  40  =  ? 
etc. 

d.   40  =  10  +  30       40  ==  60  —  20 

40  =  20  +  20  40  =  50  —  10 

50  =  10  +  40  50  =  80  —  30 

50  =  20  +  30  50  =  90  —  40 

60  =  10  +  50  30  =  70  —  40 

60  =  20  +  40       30  =  90  —  60 

60  =  30  +  30     60  =  loo  —  40 
etc. 


94  THE  SECOND   YEAR:   10-100. 


60  =  20  -f  20  +  ? 
.70  =  10  +  30  +  ? 
50  =  30  -f  10  +  ? 
80  =  40  +  10  +  ? 
90  =  70  -I-  30  —  ? 
60  =  80  +  10  —  ? 

40  =  50  +  20  —  ? 


Each  group  to  be  written  from  dictation. 

/.  Concrete  examples : 

If  a  man  travels  20  and  30  and  40  miles,  how  far  has  he 
gone  ? 

From  A  to  B  is  30  miles,  from  B  to  C  is  50  miles,  from 
C  to  D  is  20  miles.  How  far  is  A  from  D  ? 

If  I  had  90  miles  to  travel,  and  have  already  gone  30 
miles,  how  many  miles  remain  to  travel? 

(Make  more  examples  of  this  kind.) 

SECOND    UNITY. 

(Multiplication   and   Division.) 
First  Step. 

Review  first  a  few  numbers : 

1x1  =  1  i  in  1  =  1 

2x1  =  2  i  in  2  =  2 

3x1=3  i  in  3  =  3 

4x1=4  I  in  4  =  4 

till  till 

10  x  i  =  10  i  in  10  =  10 

Second  Step. 

a.  Count  off  the  first  10  miles  on  the  numeral  frame, 
also  the  2d,  3d,  etc.,  till  the  loth. 

b.  Name  the  ordinals  forwards  and  backwards  from  the 
frame  as  follows : 

That  is  the  first       10  miles  ; 

"          second  10     " 
till 

That  is  the  tenth     10  miles. 


APPENDIX.  95 

Also, 

That  is  the  tenth  10  miles  ; 
"        ninth  10     " 
etc. 

c.     i  x  10  miles  =  10  miles  ; 

2  x  10     "      =20     " 

3  x  10     "      =30     " 

4  x  10     "     =40     " 

till 
10  x  10  miles  =  100  miles. 

The  same  backwards. 
d.  Then  briefly : 

1  x  10  =  10  10  x  10  =  TOO 

2  x  10  =  20  9  x  10  =  90 

3  x  10  =  30  8  x  10  =  80 

4  x  10  =  40  7  x  10  =  70 

till  till 

10  X   10  =  IOO  I    X   10  =  10 

Third  Step. 

#.  Practise  with  concrete  and  abstract  numbers,  oral 
and  written,  without  the  numeral  frame : 

10  miles  in  10  miles  once ; 
10        "        20      "      twice;  io  =  ^of2o 

10        "       30     "      3  times;        io  =  iof3O 
till 

10        "        100    "      lotimes.       io=TVof  100,  etc. 

b.  Multiplications  and  divisions  of  abstract  and  con- 
crete numbers  not  following  regular  order : 

If  a  boy  walks  10  miles  a  day,  how  far  will  he  go  in  2 
days?  In  3,  5,  7,  6,  9,  8,  10? 

How  much  is  i  of  20  miles?  £  of  30  miles?  f  of  30 
miles?  £  of  50  miles ?  f  of  50  miles  ?  etc. 

Fourth  Step. 

Writing  and  reading  of  tens  in  both  the  form  of  mul- 
tiplication and  division. 


96  THE   SECOND   YEAR:   10-100. 

Fifth  Step. 

GENERAL  APPLICATION. 
7O  -r-  IO  =  ?     60  -5-  IO  =  ?     40  -f-  IO  =  ? 

How  many  times  10  in  30?  90?  50?  100? 

(6  x  10)  +  (4  x  10)  =  ? 

(7  x  10)  -  (5  x  10)  =  ? 

4  x  10  miles  are  how  many  more  than  2  x  10  miles? 
How  many  less  than  7  x  10  miles  ? 

How  much  is  J  of  20?  f  of  30  ?  f  of  40?  |  of  50  ? 

How  much  is  \  of  20  -f  \  of  30  +  %  of  30  ? 

How  much  is  f  of  50  —  f  of  30? 

What  number  is  3  x  10  more  than  5  x  10  ? 

A  man  has  70  miles  to  travel.  How  long  will  it  take 
him  if  he  travels  10  miles  a  day  ? 

How  many  miles  has  he  still  to  go  after  he  has  travelled 
4  days  ? 

If  a  boy  earns  10  cents  a  day  for  7  days,  how  much 
does  he  lack  of  i  dollar  ? 


SECTION  II. 

Development  of  tbe  whole  range  from  i  to  100  with  all 
numbers  between  as  sums  of  tens  and  units,  in  sec- 
tions from  i  to  20,  20  to  30,  30  to  40,  40  to  50,  50 
to  60,  60  to  80,  80  to  100.  This  includes  addition 
and  subtraction  within  these  limits. 

THIRD  UNITY.     1  to  2O. 

(Addition  and  Subtraction  of  Pure  Units  and  Tens. 
1O  +  6J  2O-5.) 

First  Step. 

Count  the  numbers  from  i  to  20  on  the  frame  forwards 
and  backwards.     Write  the  numbers  1 1  to  20. 
Children  should  be  taught — 

1.  To  distinguish  tens  and  units. 

2.  Different  values  of  figures  according  to  the  position 
or  order  in  which  they  are  found. 

3.  That  the  units  progress  just  the  same  in  connection 
with  the  tens  as  when  alone. 


APPENDIX.  97 

Second  Step. 

10  years  4-    i  year   ==  1 1  years ; 
10     "      +2  years  =12       " 
10     "      +    3     "     =  13       " 
till 

10  years  4-  10  years  =  20  years. 

20  years  —  10  years  =  10  years ; 
19      "     —    9      "     =  10 
18      "     -    8      "     =  10      " 
till 

11  years  —    i  year   =  10  years. 

i  ten  4-  i  unit  =  11 
i  "  4-  2  units  =  12 
i  "  4-  3  "  =13 

till 
i  ten  +  10  units  =  20 

11  =  i  ten  4-  i  unit. 

12  =  1    "    +2  units. 

13  =  i    "    4-3  units,  etc. 

10  4-  i  =  ii  2d—  i  =  19 

10  4-  2  =  12  20  —  2  =  IS 

10  +  3=13  20  —  3=17 

till  till 

10  4-  10  =  20  20  —  10  =  10 

Dictate  the  following  for  the  pupils  to  write  rapidly  : 
14,  ii.  17,  13,  19,  12,  etc. 

The  number  consisting  of  i  ten  and  5  units,  i  ten  and 
7  units,  i  ten  and  6  units,  etc. 

A  child  is  10  years  old.  How  old  will  he  be  in  3  years? 
In  5,  6,  9,  7,  2? 

(Other  practical  questions.) 

Third  Step. 

Separating  the  numbers  from  ii  to  20  into  tens  and 
units,  as  well  as  forming  these  numbers  from  this  and 
units.  This  must  be  oral  and  written. 


98  THE   SECOND   YEAR:    10-100. 

15  ==  i  ten  -f  5  units. 
18  =  i  ten  +  8  units. 

i  ten  -f  9  units  =  19,  etc. 

FOURTH  UNITY.     1  to  SO. 

(Addition    and    Subtraction   of   Pure  Tens   in   con- 
nection with  Mixed  Tens :     1 4  -f  1  O ;  28  -  1  O- 

First  Step. 

Count  the  numbers  from  20  to  30  forwards  and  back- 
wards on  the  frame.  Write  the  same. 

Separate  into  tens  and  units.  Form  the  new  numbers 
from  tens  and  units. 

Second  Step. 

a.  Review  the  former  exercises  oral  and  written,  and 
extend  them. 

20  miles  +  i  mile  =  21  miles; 
20    "      +2  miles  =  22      " 
20    "      +3      "     =  23 
etc. 

30  miles  —  i  mile   =  29  miles ; 
30      "     —  2  miles  =  28     " 
30     "    —  3      "     =27     " 
etc. 

6.     10  miles  4-  i  mile    =  n  miles; 

20       "        +  I        "       =  21        " 

10     "     -f  2  miles  =  12     " 

20       "        +  2        "       =  22       " 

10     "      +  3     "     =  J3     " 
20     '•      +  3      "     =  23 
etc. 

10  miles  —  i  mile  =   9  miles; 
20     "     —  i     "     =  19      " 
30     "     —  i     "     =  29 
10      "     —  2  miles  =8      " 
20     "     —  2     4<     =  18      " 

30        "       —2       "       =28         '« 

etc. 


APPENDIX,  99 

Give  also  abstract  numbers  in  irregular  order  : 

1  +  10  =  ii  11  +  10  =  21 

2  +  10=12  12+10=22 

3+  10=  13  13  +  10  =  23 

2  +  10=  12  21  —  10  =  II 

12—  10=     2  22—10=12 

13—  10=    3  23-  10=  13 

etc. 

Practise  these  exercises  and  others  until  the  pupils  are 
able  to  give  all  possible  combinations  below  30  in  adding 
and  subtracting. 

Third  Step. 

a.  Explain  the  numbers  12  and  21.  (12  =  i  ten  and  2 
units;  21  =  2  tens  and  i  unit.)  Other  numbers  in  the 
same  manner. 

b.     14  hours  +  10  hours  =  24  hours; 

18  "      +10      "      =  28      " 
7      "      +  10      "      =  17      " 

etc. 

29  hours  —  10  hours  =  19  hours  ; 

19  "  —  10   "  =9 

etc. 


20  —  8  +  10  —  20  = 
26  —  20+10+10  = 

15  +  10—  20  +  25  = 

FIFTH    UNITY. 
First  Step. 

a.  Length  of  the  schoolhouse  •  30  paces  (children  know 
as  far  as  that),  and  still  8  paces. 

Width  of  the  schoolhouse  :  16  paces. 
Other  exercises  that  require  counting. 

b.  Review  the  numbers  i  to  30. 


100  THE   SECOND   YEAR:    10-100. 

Second  Step. 

a.  Extend  the  counting  to  40  :  30  paces  +  8  paces,  etc. 

b.  Count  on  the  numeral  frame  from  30  to  40,  i  to  40, 
forwards  and  backwards. 

c.  Write  the  new  numbers  from  dictation,  and  separate 
them  into  tens  and  units. 

Third  Step. 

Exercises  like  the  previous  ones  extended  to  40, 
a.     10  +  i  =  ii        20  +  i  =  21        30  +  i  =  31 

10  +  2  =  12  20  +  2  =  22  30  +  2  =  32 

etc. 

10  —  i  =  9    20  —  i  =  19    30  —  i  =  29    40 •—  i  =  39 

10  —  2  =  8    20  —  2  =  18     30  —  2  =  28    40  —  2  =  38 

etc. 

£.10+1  10  +  2  10  +  3 

20+1         20  +  2         20  +  3 
30+1         30  +  2         30  +  3 

etc. 

10  —  I         10  —  2         10—3 
2O  —  I         2O  —  2         2O—3 

30—1      30-2      3°  -  3 
etc. 

C.   II  +  10       21+10       11+20 
12  +  10       22  +  10       12  +  20 

11  —  10       21  —  10 

12  —  IO       22  —  IO 

etc. 

d.  Mixed  exercises : 

10  +  ii  +  10  = 

20  +  ii  —  i  +  10  = 

40  —  1 1  +  6  = 

40  —  21  +  6  = 

30  —  22  +  10  =  etc. 


APPENDIX.  10 1 

Fourth  Step. 

a.  Write  12,  21,  13,  31,  23,  32. 
Separate  into  tens  and  units. 

b.  24  paces  +  10  paces  —  20  paces  = 

20  years  +  17  years  +  2  years  —15  years  = 
etc. 

Continue  until  the  new  numbers  are  thoroughly  mas- 
tered. 

SIXTH    UNITY.     1   to  5O. 
SEVENTH     UNITY.     1   to  6O. 
EIGHTH     UNITY.     1   to  SO. 
NINTH  'UNITY.     1   to    1  OO. 

The  method  pursued  in  these  will  be  the  same  as  previ- 
ously given,  and  therefore  needs  no  further  explanation. 
Dollars  and  cents  can  be  taught  in  the  last  "  unity." 


SECTION  m. 

^Development  of  the  whole  range  from  i  to  wo  by 
building  each  number  from  products  of  small  num- 
bers. This  includes  multiplication  and  division 
within  these  limits. 

TENTH     UNITY. 

(Multiplying  and  dividing  by  2.) 

First  Step. 

Children  should  be  drilled  in  the  kinds  of  coins  from  i 
dollar  down,  the  coins  being  shown  them.  They  should 
be  able  to  tell  quickly  how  many  cents  each  coin  equals  in 
value.  Attention  must  be  called  to  the  differences  be- 
tween the  coins,  so  that  the  children  can  readily  dis- 
tinguish them.  Little  examples  should  be  made,  the 
children  working  them  out  with  actual  or  toy  money  in 
their  hands.  Let  the  children  make  examples. 

Second  Step. 

a.  Count  the  cents  which  make  10  two-cent  pieces. 
Count  the  same  number  on  the  numeral  frame.  The 


102  THE   SECOND   YEAR:    10-100. 

children  arrange  the  balls  on  the  frame  in  10  twos  per- 
pendicularly under  each  other,  and  say  as  the  teacher 
points : 

That  is  the  first  two  cents ; 

"        second     " 

third 

till 

That  is  the  tenth  two  cents. 

b.  Count  2  cents,  4  cents,  6  cents  to  20  cents. 

c.  Give  the  exercises  that  follow : 

That  is  i  x  2  cents ; 
"      2x2      " 
3x2      " 

till 

That  is  10  x  2  cents. 
d.     1x2  cents  =  2  cents; 

2X2         "      =  4      " 

3x2       "    =  6     " 

etc. 

2  cents  =  1x2  cents ; 
4     "      =2x2" 
6      "      =3x2" 

etc. 
e.  Abstract : 

1X2=2  2=1X2 

2X2=4  4=2X2 

3  x  2  =  6  6  =  3x2 

till  till 

IO   X  2  =  2O  2O  =  IO   X    2 

Practise  and  write  forwards  and  backwards. 

Third  Step. 

Complete  these  numbers  with  abstract  and  concrete 
oral  and  written  examples. 

a.     1x2  cents  =    2  cents ; 
3x2     "      =6      " 
5x2     "      =10      " 
7x2     "      =14      " 
9x2     "      =  18 


APPENDIX.  103 

4  cents  =2x2  cents ; 
8     "      =    4  x  2    " 

12        "        =6X2" 

16     "      =8x2" 

2O       "        =  IO  X  2      " 

b.  2  in  2  =  once ; 
2  in  4  =  twice ; 
2  in  6  =  three  times ; 

till 
2  in  20  =  ten  times. 

i  of  4=2  2  =  £  of  4 

i  of  6  =  2  2  =  |  of  6 

t  of  8  =  2  2  =  i  of  8 
till  till 

TV  Of  20  =  2  2  =  TV  Of  20 

c.  Concrete  and  abstract  exercises  in  multiplying  and 
dividing  with  2  taken  in  irregular  order. 

Fourth  Step. 

a.  An  orange  costs  2  cents.  What  will  4  oranges  cost  ? 
8?  10? 

If  you  practise  music  2  hours  a  day,  how  many  hours 
will  you  practise  in  4  days  ?  In  a  week  ?  In  10  days  ? 

A  boy  steps  2  feet  each  time  he  steps ;  how  many  feet 
will  he  go  in  7  steps  ?  In  5  ?  In  9  ? 

(Other  examples.) 

b.  6x2  years  +  2x2  years ; 
5x2     "      +4x2     " 
4x2     "      4-3x2     " 

etc. 

10  x  2  years  —  3x2  years ; 
7x2     "      — 5x2 
8x2      "      — 2x2 
etc. 

c.  4x2  hours  +  15  hours  = 
9x2"       +12      " 
7x2"       4-10      " 

etc. 


104  THE   SECOND  YEAR:   10-100. 

8x2  hours  —  10  hours ; 

10  X  2   "    —  l6 

12  X  2   "    —  l8    " 

etc. 

d.   10  =  I  X  10       12  =  I  X  10  -f  2   14  =  I  X  10  +  4 

10  —  5X2        12  =  6X2        14  =6X2  +  2 

11  =  i  x  10  -f  i     13  =  i  x  10  +  3     15  =  i  x  10  -f  5 
11  =  5x24-1      13  =  6x2  +  1      15  =  2x6  +  3 

In  the  same  manner  to  20. 

e.     1x2  tens  =  2  tens  =  20  i  x  20  =  20 

2x2    "     =4    "     =40  2  x  20  =  40 

3x2    "     =6    "     =60  3  x  20  =  60 

4*  x  2    "     =8     "     =80  4  x  20  =  80 

5x2    "     =  10  "     =  loo  5  x  20  =  loo 

20  in  20  =  once.  20  is  i  of  40 

20  in  40  =  twice.  20  is  \  of  60 

20  in  60  =  3  times.  20  is  J  of  80 

20  in  80  =  4     "  20  is  \  of  100 
20  in  100  =5     " 

/.   2  X  II  =  22  ^  Of  22  =  II 

2  X  1 2  =  24  \  Of  24  =  1 2 

2  X  21  =  42  \  Of  42  =  21 

2  x  22  =  44  £  of  44  =  22 

(Other  numbers  in  irregular  order.) 

ELEVENTH   UNITY. 
(Multiplication   and  Division  by  3.) 
First  Step. 

a.  Count  off  threes  on  the  numeral  frame  and  arrange 
them  perpendicularly  under  each  other. 
>  Name  them  as  follows,  the  teacher  pointing  as  the 
pupil  names : 

That  is  the  first       3 ; 

second  3 ; 

third      3 ; 

till 

That  is  the  tenth     3. 


APPENDIX.  105 

b.  Count  in  intervals  of  3:    3,  6,  9,  12,  15,  to  30,  for- 
wards and  backwards. 

c.  1x3  dollars  =  3  dollars ;  3  dollars  =  1x3  dollars ; 
2x3"  =  6  6  =2x3 

3x3"       = 9      "  9      "       =3x3 

till  till 

10  x  3  dollars  =  30  dollars.  30  dollars  =  10  x  3  dollars. 

Practise  forwards  and  backwards  with  concrete  and  ab- 
stract numbers. 

Second  and  Third  Steps  by  the  same  plan  as  previously 
given. 

Fourth  Step. 

a.     i  x  3  =  3  i  x  30  =  30 

2x3  =  6  2x30=60 

3x3  =  9  3x30=90 

b.  3x1=3  3  x  ii  =  33  3  x  21  =  63  3  x  31  =  93 
3x2  =  6  3x12  =  36  3x22  =  66  3x32  =  96 
3x3  =  9  3x13  =  39  3x23  =  69  3x33  =  99 

'•  3-3=1  33-3=11  634-3  =  21  93-3  =  31 
6-5-3  =  2  36-^-3  =  12  66-5-3  =  22  96-5-3  =  32 
9-3  =  3  39-*- 3  =  13  69-5-3  =  23  99-5-3  =  33 

TWELFTH   UNITY. 
(Multiplication  and  Division  by  4.) 

Follow  the  same  plan  as  before.     In  the  last  step  in- 
troduce also : 


b. 


4 
4 
4 
4 

a 
x 

X 
X 
X 

I 

2 

3 
4 

I  X 

2  X 

3  x 
4  x 

=  4 
=  8 

=  12 

=  16 

4 
4 

4 
4 

4 
4 
4 

4 

4 
8 

12 

16 

X 

X 

x 

X 

II 

12 
13 
H 

=  44 
=  48 

-$ 

i 

2 

3 

4 

x  40 
x  40 
x  40 
x  40 

4  x 
4  x 
4  x 
4  x 

=  40 
=  80 

=  120 

=  160 

21  =84 
22  =  88 

23  =  92 
24  =  96 

J06  THE   SECOND   YEAR:    10-100. 

4-5-4=1  44-7-4=11  84-7-4  =  21 

8-7-4  =  2  48-1-4=12  88-7-4  =  22 

12-4-4  =  3  52-7-4=13  92-7-4  =  23 

16  -j-  4  =  4  56  -T-  4  =  14  96  -T-  4  =  24 

THIRTEENTH   UNITY. 
(Multiplication  and  Division  by  5.) 

Add  to  the  plan  of  the  previous  steps  : 

#.1x5  =  5  i  x  50  =  50 

2x5=  10  2x50=  loo 

3x5  =  15  3x50=150 

4x5  =  20  4x50  =  200 

5X5  =  25  5X50  =  250 

b.     5x1  =  5  5  +  11  =  55 

5x2  =  10  5  x  12  =  60 

5x3  =  15  5x13  =  65 

5  x  4  =  20  5  +  14  =  70 

5x5  =  25  5  +  15  =  75 

*.    5-4-5  =  1  55-1-5  =  11 

10-7-5  =  2  60-7-5  =  12 

15^-5  =  3  65-7-5  =  13 

20-7-5=4  70-7-5  =  14 

25-7-5  =  5  75-5  =  15 

FOURTEENTH    UNITY. 

(With  the  6.) 

FIFTEENTH     UNITY. 

(With  the  7.) 

SIXTEENTH     UNITY. 
(With  the  8.) 

SEVENTEENTH     UNITY. 
(With  the  9.) 

Nothing  further  need  be  given  to  illustrate  this  system. 
The  remaining  steps  are  carried  out  in  the  same  manner 
as  those  which  are  given  in  full, 


SECOND  COURSE 

i. 

NUMBERS  ABOVE   1OO. 


THE  THIRD  YEAR. 


FIRST  HALF  OF  THE  YEAR. 

1  OO  to    1  OOO. 

1.  As  the  numbers  between  100  and  1000  are  combina- 
tions of  the  numbers  within  the  first  hundred,  the  only 
purpose  of  this  course  is  to  reduce  them  to  their  ele- 
ments. 

2.  Thereby  the  pupil  comes  into  possession  of  the  secret 
of  all    accurate   and  rapid    mental  work   in   arithmetic, 
namely,  always  to   operate  with   the    smallest    possible 
numbers;  hence  he  needs  none  of  the  so-called  "arith- 
metical knack." 

3.  In  order  to  lead  to  an  allsided  representation  of  the 
number,  it  is  impossible  to  consider  all  of  the  fundamen- 
tal rules  at  once  as  heretofore ;  this  will  receive  wider  at- 
tention in  the  second  half  of  the  year.  Mental  and  written 
arithmetic  are  now  united  at  every  step. 

4.  As  the  necessity  of  isolating  each  number  now  dis- 
appears, and  because  the  allsided  penetration  and  com- 
prehension of  each  number  must  take  place,  the  material 
will  be  divided  in  two  parts  only : 

A.  The  pure  number :  measuring,  separating,  compar- 
ing, and  combining. 

B.  The  applied  Number. 


IOS  THE    THIRD   YEAR:   IOCKIOOO. 

The  child  is  now  sufficiently  mature  gradually  to  leave 
simple  mechanical  processes  and  make  more  use  of  the 
understanding  and  reason.  He  must,  however,  master 
the  processes,  so  as  to  be  able  to  give  them  rapidly  and 
almost  mechanically.  This  side  of  the  work  is  not  to  be 
neglected.  Illustrations  should  still  be  used  where 
needed,  but  he  will  learn  chiefly  by  analogy  from  smaller 
numbers. 

A. — Allsided  Contemplation  of  the  Pure  Number. 

(First  Quarter.) 

FIRST  STEP. 

Measuring  of  the  numbers  by  the  units  of  the  Decimal 

System,  by  units,  tens,  and  hundreds. 
*  a.  (Oral.) 

Count  upwards  and  downwards  from  100  to  1000.  10 
splints  can  be  bound  together,  and  that  is  i  ten.  Around 
10  of  these  bundles  (100)  a  wide  ribbon  may  be  tied  ;  10  of 
these  bundles  make  1000.  In  this  way  the  pupil  gains  a 
comprehension  of  1000.  Solid  blocks  divided  by  lines 
into  10  and  100  units  can  also  be  used. 

During  the  counting  the  pupils  must  often  be  stopped 
and  questioned  as  to  which  hundred  and  which  ten  they 
are  in  :  How  many  units  and  tens  are  lacking  in  the  tens 
and  hundreds  respectingly,  and  in  the  hundreds  how  many 
are  lacking  from  a  thousand  ? 

For  example  :  The  teacher  gives  768.  The  pupil  will 
explain : 

768  =  7  hundreds,  6  tens,  8  units.  It  lacks  2  units  of 
completing  the  7th  ten,  then  3  tens  of  completing  the  8th 
hundred,  and  finally  2  hundreds  of  completing  1000. 

829  ==  8  hundreds,  2  tens,  9  units.  Complete  the  analysis 
of  829. 

Analyze  in  the  same  way  999,  500,  463,  271,  604. 

What  number  is  composed  of  3  hundreds,  6  tens,  and 
5  units? 

How  many  units  in  7  hundreds,  8  tens,  and  9  units  ? 

How  many  units  in  one  thousand  ?   Tens  ?    Hundreds  ? 

Of  what  does  669  consist  ? 

(6  x  100)  +  (6  x  10)  +  (9x1) 


FIRST  HALF  OF   THE   YEAR.  IOQ 

b.  (Written.) 

To  make  it  easier  for  beginners  in  writing,  use  the  fol- 
lowing plan : 

h.  t.  u. 

i  o  i  =  101 

4  8  o  =  480 

10  o  o=  1000 

Numbers  must  be  dictated  for  the  pupils  to  write.  When 
written  on  the  blackboard,  the  figures  should  be  named 
and  the  numbers  read,  etc.,  in  order  to  acquire  perfect 
mastery  of  the  subject. 

Finally,  analogous  to  the  oral  work,  dictated  numbers 
should  be  written  out  as  follows  : 

615  =   6  x  100  +  i  x  10  +  5  x  i 

2O4  =     2  X   ICO  -fOX   IO  +  4  X   I 

390  =    3  x  loo  +  9  x  10  +  o  x  i 

1000  =  10  X   IOO  +  0X10  +  0X1 

or 
615  =  600  +10  +  5 

204  =  200  +  4 
390  =  300  +  90 

IOOO  =  IOOO 

In  the  following  steps  we  shall  give  only  one  form, 
which  will  answer  for  both  oral  and  written  work. 
Neither  is  to  be  omitted,  but  they  must  be  united,  the 
oral  taking  precedence  in  order  of  time.  Follow  the  plan 
suggested  in  the  First  Step. 

SECOND  STEP. 
The  pure  hundreds  measured  with  hundreds. 

Measuring,  comparing,  rapid  work,  combining,  are  to  be 
the  same  as  in  the  First  Course.  With  the  number  2  in 
the  First  Course  we  obtained  the  following  scheme : 

1  +  1=2 

2X1=2 

2—1  =  1 

2  -f-  I  =  2 


HO  THE    THIRD   YEAR:    100-1000. 

Conformably  to  this  the  pupil  now  learns  : 
200. 

100  +  100  =  200 
2  X  IOO  =  2OO 

2OO  —  IOO  =  IOO 
200  -T-  IOO  =  2 

What  number  is  contained  twice  in  200? 
Of  what  number  is  200  the  double  ? 
Of  what  number  is  100  the  half? 
What  number  must  I  double  in  order  to  get  200  ? 
(Give  many  other  examples,  following  the  method  em- 
ployed in  teaching  the  2.    See  Second  Step  in  the  First 
Course.) 

300. 

100  -f  loo  +  ioo  =  300 
3  x  ioo  =300 
300  -i-  ioo  =  3 

200  +  IOO  =  300 
3OO  —  IOO  =  2OO 
3OO  —  2OO  =  IOO 

300  -f-  200  =  i,  with  remainder  of  ioo. 

300  is  ioo  more  than  200,  200  more  than  ioo. 

200  is  ioo  less  than  300,  ioo  more  than  a  hundred. 

ioo  is  200  less  than  300,  ioo  less  than  200. 

300  is  3  times  ioo. 

ioo  is  \  of  300. 

Of  what  equal  and  what  unequal  numbers  does  300 
consist  ? 

How  much  is  300  —  ioo  —  ioo  +  200  ? 

300  •«-  3  —  ioo  -f  200  -f-  ioo  x  ioo? 
300  —  200  +  ioo  +  ioo  -5-  3  —  ioo  ? 

From  what  number  can  you  take  away  2  x  ioo  and 
have  ioo  left  ? 

£  of  300  is  how  much  less  than  \  of  300  ? 
Which  is  greater,  \  of  300  or  \  of  200  ? 


FIRST  HALF  OF  THE  YEAR.       1 1 1 
400. 


I.  Measure  with  100 : 


100  +  100  +  100  -f  100  =  400 
4  x  100  =  400 

400  —  100  —  100  —  100  =  100 
400  -f-  100  =  4 


2.  Measure  with  200 : 


200  4-  200  =  400 
2  x  200  =  400 
400  —  200  =  200 

400  -5-  200  =  2 

3.  Measure  with  300 : 

300  +  100  =  400 
100  +  300  =  400 
i  x  300  +  100  =  400 
400  —  200  =  200 
400  —  300  =  100 
400  -5-  300  =  i  (100) 

400  is  100  more  than  300 
200  more  than  200 
300  more  than  100 

300  is  100  less  than  400 
100  more  than  200 
200  more  than  100,  etc. 

THIRD  STEP. 
Mixed  hundreds  measured  mith  mixed  hundreds. 

220  =  2  x  no,  also  i  x  220 
440  —  4  x  no,  also  2  x  220 
660  =  6  x  no,  also  3  x  220 

880  =  8  x  1 10,  also  4  x  220,  2  x  440 
990  =  9  x  1 10,  also  3  x  330 


112  THE    THIRD   YEAR:    100-1000. 

How  may  888,  999,  be  considered  ? 

888  =  8  x  in,  also  4  x  222,  2  x  444 
999  =  9  x  in,  also  3  x  333 

999  •*•  333  =  888  +•  222  = 

999  -*-  3  =  888  -5-  4  = 

999  -f-  1 1 1  =  888  -H  8  = 
etc.  etc. 

Of  what  number  is  120  the  J,  £,  i? 
What  is  i  of  844  ? 
Of  what  number  is  844  fourfold? 
What  number  can  I  take  4  times  from  844  ? 
What  number  is  contained  4  times  in  844? 
\  of  844  is  how  much  greater  than  J  ? 
i  of  333  is  -J-  of  what  number? 

Of  what  number  is  \  of  333  the  ninth?   (\  of  333  =  in. 
in  is  \  of  9  times  1 1 1  =  999.) 
Compare  365  with  244. 

365  =  3  hundreds  4-  6  tens  +  5  units ; 
244  =  2  hundreds  +  4  tens  +  4  units. 

3  hundreds  —  2  hundreds  =  i  hundred ;  6  tens  —  4  tens 
=  2  tens  ;  5  units  —  4  units  =  i  unit.  Therefore  365  — 
244  =  i  h.+  2  t.-f  i  u.=  121,  or  365  is  121  greater  than 
244,  and  244  is  121  less  than  365. 

What  is  the  difference  between  743  and  120? 

743  =  7  h.-f  4t.+  3u.;  120=  i  h.+  2  t.  +  ou.;  7  h.  - 
i  h.  =  6h.  4t.  —  2t.  =  2  t.;  3u.— ou.  =  3  u.  Therefore 
743  —  120  =  6  h.  +  2  t.  +  3  u.  =  623. 

What  number  =  743  +  221  ? 

743  =  7  h.+  4t.+  3  u.;  221  =2  h.+  2  t.+  i  u.;  7  h.  +  2 
=  9  h.;  4  t.  +  2  t.  =  6  t.;  3  u.  +  i  =  4  u.  Therefore  743  + 
221  =  9  h.  -I-  6  t.  +  4  u.  or  964. 

How  much  is  in  -f  212  +  313? 

How  much  is  112  +  113  -f  114? 

Subtract  322  and  124  from  659. 

Continue  these  exercises  until  the  subject  is  undei- 
stood. 


FIRST  HALF  OF   THE   YEAR.  1 13 

FOURTH  STEP. 
Measuring  of  hundreds  with  tens. 

I. 

a.  The  pure  hundreds. 
Since  100  =  10  x  10, 

2  x  100  or  200  =  2  x  10  x  10  =  20  x  10 

3  x  100  or  300  =  3  x  10  x  10  =  30  x  10 

4  x  loo  or  400  =  4  x  10  x  10  =  40  x  10 

10  x  loo  or  looo  =  10  x  10  x  10  =  100  x  10  =1000. 

b.  Hundreds  with  tens. 
Since  100  =  10  x  10, 

no  =  (10  x  10)  +  (i  x  10)  =  ii  x  10 

120  =  (10  X  10)  +  (2  X  10)  =  12  X  10 

130  =  (to  x  10)  +  (3  x  10)  =  13  x  10 

140  =  (10  x  10)  +  (4  x  10)  =  14  x  10 

150  =  (10  x  10)  +  (5  x  10)  =  15  x  10 

190  =  (10  x  10)  +  (9  x  10)  =  19  x  10 

240  =  (20  x  ip)  +  (4  x  10)  =  24  x  10 

990  =  (90  x  10)  +  (9  x  10)  =  99  x  10 

c.  Hundreds  with  tens. 
Since  100  =  10  x  10, 

ioi  =  (10  x  10)  +  i 
109  =  (10  x  10)  +  9 
906  =  (90  x  10)  -f  6 
814  =  (81  x  10)  +  4 

How  many  tens  in  500,  900,  1000? 
What  number  =  53  tens  ? 
What  number  =  9  units  more  than  53  tens  ? 
How  many  times  10  is  660,  420,  870  ? 
Of  what  number  is  10  the  42d  part  ?     The  66th  ?    The 
84th  ?    The  ;oth  ? 

How  many  tens  in  879? 


114  THE    THIRD   YEAR:    100-1000. 

II. 

Comparison. 

Compare  400  with  900. 

(400  =  40  tens ;  900  =  90  tens ;  90  tens  —  40  tens  =  50 
tens.  Therefore  900  has  50  tens  more  than  400,  and  400 
has  50  tens  less  than  900.) 

How  many  are  55  tens  less  than  600?  Than  660? 
Than  990? 

(As  600  =  60  tens,  55  tens  are  5  tens,  or  50  less  than 
600.) 

Of  what  4  equal  tens  does  880  consist  ? 

(880  =  88  tens,  and  as  88  tens  =  4  x  22  tens,  880  is 
composed  of  4  x  22  tens.) 

What  is  the  sum  of  800,  180  and  20? 

(800  +  180  -f  20  =  80  +  18  +  2  tens  =  100  tens  =  1000.) 

What  is  the  difference  between  160  and  210? 

(210  or  21  tens  —  160  or  i6tens  =  5  tens  =  50.) 

60  x  10  =  how  many  times  100  ? 

What  number  has  8  tens  and  9  units  more  than  490? 

(The  number  which  has  8  tens  and  9  units  more  than 
490  must  equal  490  +  8  tens  +  9  units.  490  or  49  tens  + 
8  tens  =  57  tens  =  570.  570  -f  9  =  579.) 

I  have  taken  a  number  87  times,  added  9  to  it  and  ob- 
tain 879.  What  is  the  number  ? 

(879  =  87  tens  +  9  units.  Therefore  I  must  have  taken 
ten  87  times.) 

How  many  more  tens  has  73  x  10  than  the  double  of 
240? 

How  many  times  is  yj^  of  1000  contained  in  500? 

(looo  =  loo  tens  ;  yj-g-  of  100  tens  =  i  ten  ;  500  =  50 
tens ;  50  tens  contains  i  ten  50  times.) 

\  of  630  =  J  of  what  number  ? 

(630  —  63  tens ;  i  of  63  tens  =  21  tens ;  21  tens  is  i  of  4 
x  21  tens  =  84  tens  =  840.) 

-5*5-  of  680  +  ^  of  240  is  how  much  less  than  10  x  36  ? 

(As  680  =  68  tens,  ¥V  of  680  =  i  ten  ;  and  -fa  of  240  = 
I  ten  ;  both  together  =  2  tens.  10  x  36  =  36  tens,  and  36 
tens  —  2  tens  =  34  tens.) 

The  factors  must  also  be  changed  in  these  exercises. 


FIRST  HALF  OF   THE   YEAR.  11$ 

110=  II    X    10=  10   X    II 
22O  =  22  X   IO  =  IO  X   22 

680  =  68  x  10  =  10  x  68 
990  =  99  x  10  =  10  x  99 

Of  what  is  990  composed  ? 
How  can  130  be  composed  from  13  ? 
280  from  28  ?     560  from  56  ? 
What  number  must  I  take  10  times  to  get  670? 
Of  what  number  is  67  one  tenth  ? 
What  is  -gV  of  670  ? 

How  many  times  is  79  contained  in  790  ? 
What  number  can  I  take  10  times  from  790  ?     What  79 
times  ? 
79  x  10  =10  times  what  number  ? 

These  exercises  lead  us  naturally  to  the  next  step. 

FIFTH  STEP. 
Measuring  a  number  by  its  factors. 

I. 

a.  The  pure  hundreds. 

100  =  2  x  50,  4  x  25,  5  x  20,  10  x  10 


Therefore, 

200  = 

2 

x 

2 

x 

50 

= 

4 

X 

50 

2 

x 

4 

X 

25 

= 

8 

x 

25 

2 

X 

5 

X 

20 

= 

IO 

X 

20 

2 

X 

10 

X 

10 

— 

20 

x 

10 

300  = 

3 

X 

2 

X 

50 

= 

6 

X 

50 

3 

x 

4 

X 

25 

= 

12 

X 

25 

3 

X 

5 

X 

20 

= 

15 

X 

20 

3  x  10  x  10  =  30  x  10  =  20  x  15 
b.  Hundreds  with  tens. 

220  =  10  X   22.      As     10  =  2   X    5,    22O  =  2X5X22  =  2 

x  1 10 ;  and  as  22  =  2  x  1 1,  10  x  2  x  1 1  =  10  x  22 ;  and 
as  10  =  5  x  2,  5  x  2  x  22  =  5  x  44. 


Il6  THE    THIRD   YEAR:    100-1000. 

960  =  10  x  96. 

10   X   2  X  48  ==  20   X  48  =  48   X   20 

10  x  3  x  32  =  30  x  32  =  32  x  30 
10  x  4  x  24  =  40  x  24  =  24  x  40 
10  x  6  x  16  =  60  x  16  =  16  x  60 

10  X   8   X    12  =  80   X    12  =  12   X   80 

Or  leaving  the  second  factor  unchanged : 

iox96  =  2X5X96  =  2x  480 
$X2X96  =  5x  192 

c.  Hundreds  with  tens  and  units. 

426  =  (10  x  42)  +  6  or  (4  x  100)  -f  26 

(10  x  2  x  21)  +  6  or  (20  x  21)  +  6 

(10  x  3  x  14)  +  6  or (30  x  14)  +  6  =  (i4x  30)4-6 

(2  x  5  x  42)  -f  6  =  (2  x  210)  +  6 

(5  x  2  x  42)  +  6  =  (5  x  84)  +  6 

896  =  (8  x  100)  +  (8  x  12)  =  8  x  112 

(8  XII 2)  =    2X4X1 12  =  2X  448 

4x2  x  ii2=4x  224 

(10  x  89)  +  6  =  (2  x  5  x  89)  +  6  =  (2  X445)  +  6 
(5  x  2  x  89)  +  6  =  (5  x  178)  +  6 

489=3(10  X48)  +  9 
(2  x  240)  +  9 

(5  x  96)  +  9 
Or: 

(10  x  4  x  12)  +  9  =  (40  x  12)  +  9  =  (12  x  40)  +  9 
(10  x  3  x  16)  +  9  =  (30  x  16)  +  9  =  (16  x  30)  +  9 
(10  x  2  x  24)  +  9  =  (20  x  24)  +  9 

300  is  composed  of  how  many  twos  ?    Threes  ?    Fives  ? 

How  do  you  find  the  25th  part  of  300  ? 

(^5  of  100  =  4,  so  ^V  of  300  =  3  x  4  =  12.) 

How  does  300  arise  out  of  15  ? 

(As  300  =  10  x  30,  and  30  =  2  x  15,  300  =  10  x  2  x  15 
=  20  x  15.  I  have  taken  15  twenty  times  and  obtained 
300.  Or:  as  300  =  2  x  150,  and  150=  10  x  15,  300  =  2 
x  10  x  15  =  30  x  15.) 

How  many  times  must  I  take  44  to  get  220  ? 

(220  =IOX22=5X2X22=5X  44.) 


FIRST  HALF  OF   THE   YEAR.  1 1/ 

What  number  must  I  take  5  times  from  426  in  order  to 
have  6  remainder  ? 

(426  =  10  x  42)  +  6  =  (5  x  2  x  42)  -f  6  =  (5  x  84)  +  6. 
So  I  must  subtract  84  five  times  from  426,  and  have  6 
remainder.) 

How  many  times  is  24  contained  in  489? 

(489  =  [10  x  48]  +  9  =  [10  x  2  x  24]  +  9  =  [20  x  24] 
+  9.  24  in  489  20  times  with  9  remainder.) 

II. 

Comparison. 

What  is  the  difference  between  980  and  377? 

(980  =  98  tens,  and  377  =  37  tens  and  7  units.  98  tens 
—  37  tens  =  61  tens  —  7  units  =  60  tens  +  3  units  =  603. 
Or  :  900  —  300  =  600 ;  80  —  77  =  3 ;  980  —  377  =  603.) 

The  difference  between  980  and  377  is  3  times  what 
number  ? 

(The  difference  between  980  and  377  is  603.  603  is  3 
times  i  of  603.  \  of  600  =  200,  \  of  3  =  i,  200  +  i  =  201. 
Therefore,  etc.) 

•J-  and  £  of  480  taken  together  is  how  many  less  than 
twice  480  ? 

By  what  number  must  I  divide  365  to  get  5  ? 

(If  I  divide  365  by  a  number  and  get  5,  that  number  is 
contained  5  times  in  365,  or  is  \  of  365.  \  of  300  =  60 ;  \ 
of  65  =  13  ;  60  -f  13  =  73.  Therefore,  etc.) 

What  is  the  difference  between  ^  and  ^  of  660  ? 

The  sum  of  326  and  418  is  how  much  greater  than  the 
sum  of  their  halves  ? 

I  take  4  units  from  a  number,  and  then  divide  the  re- 
mainder by  1 6,  and  obtain  a  quotient  of  60.  What  is  the 
number  ?  % 

(The  unknown  number  is  16  x  60  +  4.  60  =  6  tens; 
1 6  x  6  tens  =  96  tens  =  960;  960  +  4  =  964.  Therefore, 
etc.) 

What  number  is  10  more  than  the  double  of  5  x  99? 


Il8  THE    THIRD   YEAR:   100-1000. 

SIXTH  STEP. 
Reduction  of  numbers  from  i  to  1000  to  their  elements. 

It  does  not  matter  in  what  order  the  numbers  are  taken, 
the  chief  object  being  practice  in  rapidly  and  accurately 
separating  the  numbers  into  their  elements.  The  pupil  is 
now  able  to  tell  at  a  glance  into  what  parts  the  number 
must  be  separated.  The  teacher  should  make  the  work 
partly  oral  and  partly  written. 

3GO. 

300  +  60  3  x  loo  +  3  x  20 
i  So  -f  1 80  3  x  1 20 

200  4-  160  10  x  36 
320  +  40  5  x  72 

336  +  24  20  x  18  =  18  x  20 

etc.  9  x  40  etc. 

320  +  45,  2  x  150  -f  65,  2  x  182  +  i,  7  x  50  +  15,  14  x 
25  +  15,  18  x  20  -f  5,  etc. 

These  six  steps  complete  the  work  of  the  first  quarter. 
The  work  now  changes  from  the  pure  to  the  applied 
number.  At  this  point  especial  attention  should  be  given 
to  compound  numbers,  weights,  measures,  money,  etc. 
This  will  employ  the  second  quarter.  The  teacher  must 
supply  a  great  many  more  examples  than  are  here  given, 
as  the  "  applied  number"  is  of  great  importance. 

B. — All-sided  Contemplation  of  the  Applied  Number. 
(Second  Quarter.) 


a.  The  tens. 


10  cents    =  i  dime. 
10  dimes  =  i  dollar. 
10  dollars  =  i  eagle. 

3  dimes   =  30  cents. 

5  dimes   =  50  cents  or  \  dollar. 
50     "        =5  dollars  or  \  eagle. 


FIRST  HALF  OF  THE  YEAR.        I  1 9 

1  dollar  —  10  dimes. 

2  dollars  =  20 

3  "       =30 
5      "       =50 

10         "  =  100 


100  cents    =  10 


or  i  dollar. 


loo  dollars  =  1000  dimes. 
900  dimes   =  30  x  30  dimes. 
870      "        =  29  x  30      " 
840      "        =  28  x  30 
810      "        =  27  x  30      " 
etc. 

9,  n,  17,  28  dollars  =  how  many  dimes? 

9  dollars,  4  dimes,  24  cents  =  how  many  cents  ? 

314,  365,  720,  799  cents,  how  many  dimes?  How  many 
dollars  ? 

25  dimes  +  9  dimes  -f  17  dimes  +  15  dimes  =  how 
many  dimes  ?  How  many  dollars  ? 

From  2  dollars  and  6  dimes  take  15  dimes.  17  dimes. 
25  dimes. 

Divide  the  class  into  two  divisions  and  let  the  first 
division  give  3  times,  and  the  second  division  4  times,  the 
numbers  as  the  teacher  names  them. 


Teacher. 

ist  Division. 

2d  Division. 

25  dimes. 

75  dimes. 

100  dimes. 

9      " 

27      " 

36      « 

15    « 

45 

60      " 

17    " 

5i      " 

68      •« 

Again,  the  first  division  can  give  3  times  the  number 
given,  and  the  other  division  add  the  results. 

Teacher.  ist  Division. 

19  dimes.     5  dollars  and  70  cents. 
12      "          3       "        "    60      " 
22      "          6       "        "     60 

2d   Division 15       "        "    90      " 

In  long  examples  the  results  of  multiplying  can  be  writ- 
ten in  order  to  get  the  total  correctly. 


120 


THE    THIRD   YEAR:  100-1000. 


14  dollars  9  dimes. 


3 

17 
18 

19 

102 


7 
4 
8 
6 
5 

9 


This  will  be  added  as  follows,  the  operations  being 
mental,  only  the  above  being  set  down  : 

14  dollars  9  dimes  +  15  dollars  =  29  dollars  9  dimes,  -f 
7  dimes  =  30  dollars  6  dimes;  30  dollars  6  dimes  +  16 
dollars  4  dimes  =  47  dollars;  47  dollars  -f  17  dollars  8 
dimes  =  64  dollars  8  dimes;  64  dollars  8  dimes  4-  18  dol- 
lars 6  dimes  =  83  dollars  4  dimes ;  83  dollars  4  dimes  + 
19  dollars  5  dimes  =  102  dollars  9  dimes. 

2.  Reverse  the  process,  subtracting  19  dollars  5  dimes 
from  102  dollars  9  dimes,  and  so  on  until  the  first  number, 
14  dollars  9  dimes,  is  reached. 

3.  Multiply  each  of  these  numbers  by  2,  3,  4,  5,  and  add 
the  products,  and  divide  the  sum  by  2,  3,  4,  5,  and  see  if 
the  sum  of  the  original  numbers  is  obtained. 

14  dollars  9  dimes  x  3  =  44  dollars  7  dimes. 


II 

17 
18 

12- 

102 


7 
4 
8 
6 
5 


47 
49 
53 
55 
J8 

308 


3  in  308  dollars  7  dimes  =  102  dollars  9  dimes. 

b.  Tens  and  units. 

24  dimes    =  how  many  cents  ? 
24  dozen    =     "        "      things  ? 
24  dollars  =     "        "      dimes  ? 


FIRST  HALF  OF  THE  YEAR. 


121 


Pupils  must  be  drilled  in  tables  from 

12  x  i  to  24  x  12, 

15  x  i  to  15  x  15, 

16  x  i  to  16  x  16, 
24  x  i  to  24  x  24. 

One  dollar  =  10  dimes  =  100  cents. 
One  half      "      =    5 
One  fourth       "      =    2^ 

One  fifth       "      =    2  =20 

One  tenth       "      =    i  =    10 

One  twentieth  =     £  =5 

One  fiftieth       "      =     |  =2 

One  hundredth      "      =  -^  =      i  cent. 

100  cents  =  10  =      i  dollar. 
So     "      =    5 

20  =     2£ 

etc. 

The  English  weights  and  measures  cannot  well  be  ap- 
plied to  the  Grube  system,  as  the  scales,  instead  of  being 
decimal,  are  varying.  And  yet,  with  their  present  knowl- 
edge of  the  numbers  as  high  as  1000,  and  with  the  appli- 
cations of  compound  numbers  made  in  the  First  Course 
during  the  first  two  years,  the  pupils  should  be  able  to 
master  the  subject  of  compound  numbers  in  connection 
with  the  above  work  during  the  second  quarter. 


II. 


THE  FOUR  FUNDAMENTAL  RULES  IN 
ABSTRACT  AND  CONCRETE  NUMBERS 
-UNLIMITED  RANGE. 


THE  THIRD  YEAR. 

SECOND  HALF  OF  THE   YEAR. 

The  division  of  the  work  is  as  follows  : 
A. — With  abstract  numbers. 

!i.  Numeration. 

2.  Addition. 

3.  Subtraction. 

4.  Multiplication. 

[  5.  Division. 

B. — With  concrete  numbers. 

f  i.  Numeration. 

12.  Addition. 

3.  Subtraction. 

4.  Multiplication. 

5.  Division. 

Mental  and  written  arithmetic  must  not  be  separated, 
as  though  they  were  different  studies ;  mental  arithmetic 
is  the  foundation  ;  here  the  idea  has  its  beginning;  on  this 
the  written  statement  depends. 

It  becomes  necessary  in  large  numbers,  and  where 
several  numbers  are  involved,  to  write  them  down  in 


SECOND  HALF  OF    THE   YEAR.  123 

figures,  so  that  they  can  be  seen  by  the  eye.  The  writ- 
ten work  may  thus  be  considered  as  only  an  assistant  to 
the  mental  work. 

It  must  not  be  thought  that  in  addition  there  shall  be 
only  adding,  and  in  subtraction  only  subtracting ;  but,  as 
in  the  preceding  steps,  all  the  operations  must  be  united 
as  far  as  may  be.  Many  examples  must  be  given.  The 
exercises  in  "  rapid  work"  must  not  be  neglected,  and 
emulation  among  the  pupils  must  be  aroused  to  secure 
rapid  and  accurate  work. 

Only  the  important  points  of  each  step  will  be  given, 
the  filling  in  with  material  being  left  to  the  skill  and  tact 
of  the  teacher. 

A. — With  Abstract  Numbers. 

FIRST    STEP. 

Numeration. 

a.  (Oral.)  Thousands  and  millions.  If  we  have  the  10 
hundreds  we  have  a  new  unity — the  thousand.  We  can 
combine  thousands  into  tens  of  thousands ;  tens  of  thou- 
sands into  hundreds  of  thousands,  etc.,  exactly  as  we 
combined  units  into  tens,  tens  into  hundreds,  etc. 

unit  =    i  unit  =  i. 

unit  =  10  units  =  10. 

ten  =  10  tens  =  100. 

hundred    =  10  hundred  =  i  thous.  =  1000. 

thousand  =  10  thousand  =  i  tenth.  =  10000. 

ten  thous.  =  10  ten  thous.  =  i  h'd  th.  =  100000. 

hund.  th.  =  10  hund.  thous.  =i  million  =  loooooo. 


i  x 
10  x 
10  x 

JO  X 

10  x 
10  x 
10  x 


In  the  same  way : 


i  x  2  units  =  2  units  —  2 

10   X   2         "       =  20     "       =  20 

10  x  2  tens    =  20  tens  =  200 
etc. 

i  x  3  units  =  3  units  =  3 
10  x  3      "     =  30    "     =  30 
10  x  3  tens    =  30    "     =  300 
etc. 


I24 


THE    THIRD   YEAR:   100-1000. 


i  unit  —    i  x  I  unit 

i  ten  =  10  x  i  unit          =  10  units. 

i  hundred  =  10  x  i  ten          =  100  units. 
i  thousand  —  10  x  i  hundred  =  100  x  i  ten  =  1000  units. 
etc. 

The  units  are  units  of  the  first  order,  the  tens  .are 
units  of  the  second  order,  the  hundreds  are  units  of  the 
third  order,  etc.  The  units  of  each  order  are  10  times 
more  than  the  units  of  the  preceding  order.  In  each 
order  there  can  be  only  10  units,  and  the  tenth  unit  is  the 
first  of  the  following  order. 

b.  (Written.) 

We  will  write  the  number  1852,  placing  the  units,  tens, 
hundreds,  and  thousands,  each  in  a  separate  box. 


Counting  from  the  right  towards  the  left,  we  have 
units  in  the  first  box,  tens  in  the  second,  hundreds  in  the 
third  and  thousands  in  the  fourth,  tens  of  thousands  in 
the  fifth,  hundreds  of  thousands  in  the  sixth,  millions  in 
the  seventh. 


Or, 


Read'.        6  millions. 

0  hundred  thousands. 
9  ten  thousands. 

2  thousands. 

1  hundred. 
6  tens. 

o  units. 

6  million  92  thousand  i  hundred  and  60. 


Give  the  name  of  the  units  of  the  ist,  3d,  5th,  4th,  6th 
order. 
What  are  units  of  the  8th  order  called  ?  etc.,  etc. 


SECOND  HALF  OF   THE    YEAR. 


12$ 


Then  the  teacher  writes  9  in  each  box  successively,  and 
the  children  read  it. 


8 

7 

6 

5 

4 

3 

2 

I 

9 

9 

o 

9 

o 

0 

9 

o 

o 

o 

9 

o 

0 

0 

o 

9 

o 

o 

o 

o 

0 

9 

o 

0 

o 

o 

o 

o 

9 

o 

o 

0 

0 

0 

o 

o 

Again  the  teacher  writes  and  the  children  read : 

90,000,000 

9,000,000 

900,000 

90,000 

9,000 

900 

90 

9 

Instead  of  boxes  columns  may  be  used  as  follows. 
The  teacher  writes,  and  the  children  rea 


126 


THE    THIRD   YEAR:   loo-iooo. 


a. 

Mil. 

8 
T. 

Millions 

7 
Millions 

6 
H. 

Thous. 

4. 

Thous. 

4 

Thous. 

3 
Hund. 

2 

Tens. 

X 

Units. 

3 

O 

S 

6 

4 

8 

7 

0 

O 

2 

6 

5 

9 

O 

8 

4 

7 

0 

o 

9 

3 

6 

4 

2 

i 

9 

8 

6 

O 

The  pupils  should  arrange  such  columns  on  their  slates, 
and  on  the  blackboard  then  write  numbers  in  them  from 
dictation.  They  must  also  be  drilled  in  such  exercises  as : 

What  is  the  3d  order,  the  5th,  the  8th,  the  6th,  the  4th, 
etc.  ? 

The  hundreds  are  what  order  ?  The  millions  ?  The 
tens  of  thousands  ? 

The  orders  must  be  known  from  right  to  left  and  left 
to  right,  so  as  to  be  given  without  hesitation  with  great 
facility. 

Then  the  numbers  may  be  divided  into  periods  (threes), 
attention  being  called  to  the  fact  that  there  are  "units," 
"  tens,"  and  "  hundreds"  of  units ;  also  units,  tens,  and 
hundreds  of  thousands,  millions,  etc. 

The  periods  may  be  shown  as  follows : 


Million.       Thousand. 


Units. 


h. 
3 

t. 
2 

u. 

5 

h. 

I 

t. 
7 

u. 
6 

h. 

I 

t. 
8 

u. 
0 

Then  all  columns,  etc.,  may  be  abandoned,  and  the 
numbers  written  in  periods.  Always  have  the  commas, 
which  separate  the  periods,  placed  in  at  the  time  of  writ- 
ing the  number,  and  not  after  all  the  figures  have  been 
written.  Thus:  325  million  (comma),  176  thousand 
(comma),  180.  Thus  the  pupil  becomes  perfectly  sure  in 
both  writing  and  reading  numbers. 


SECOND  HALF  OF  THE  YEAR.          127 

SECOND  STEP. 

Addition. 
ORAL  AND  WRITTEN. 

a.  Numbers  of  one  figure  : 

4  units  -I-  5  units  =  9  units  (4  +  5  =  9). 
4  tens  +  5  tens  —  9  tens  =  90  (40  -f  50  =  90). 
4  hund.  +  5  hund.  =  9  hund.  =  900  (400  4-  500  =  900)0 
4  th.  +  5  th  =  9  th.  =  9,000  (4,000  +  5,000  =  9,000). 
4  ten  th.  +  5  ten  th.  =  9  ten  th.  =  90,000  (40,000  +  50,000 
=  90,000). 

etc. 

b.  Numbers  of  two  figures  and  one  figure  : 

43  units  -f  5  units  =  48  units  (43  +  5  =  48). 
43  tens  +  5  tens  =  48  tens  =  480  (430  +  50  =  480). 
etc. 

c.  Numbers  of  two  figures  and  two  figures : 

43  units  -f  28  units  =  71  units  (43  +  28  =  71). 
43  tens  +  28  tens  =  71  tens  =  710  (430  -f-  280=  710). 
etc. 

d.  Numbers  of  three  figures  and  one  figure  : 

416  units  +  8  units  =  424  units  (416  +  8  =  424). 
416  tens  +  8  tens  =  424  tens  =  4,240  (4,160  +  80  =  4,240). 
etc. 

e.  Numbers  of  three  figures  and  two  figures  : 

416  units  +  23  units  =  439  units  (416  +  23  =  439). 

416  tens  +  23  tens  =  439  tens  =  4,390  (4,160  +  230  = 
4,390)- 

416  hund.  +  23  hund.  =  439  hund.  =  43,900  (41,600  -f 
2,300  =  43,900). 

416  th.  +  23  th.  =  439  tn-  =  439,000  (416,000  +  23,000 
=  439,000). 

etc. 


128  THE    THIRD   YEAR:    100-1000. 

/.  Numbers  of  three  figures  and  three  figures  : 

416  units  4-  123  units  =  539  units  (416  +  123  x  539). 
416  tens  +  123  tens  =  539  tens  =  5,390  (4,160  +  1,230 

=  5»39°)- 

416  hund.  4-  123  hund.  =  539  hund.  =  53,900  (41,600  4- 
12,300=53,900). 

etc. 

The  corresponding  method  for  operations  to  acquire 
rapidity.     For  examples : 

a.  7         4-8         +9         +6 
70       +80       +90       +60 
700      4-  800      4-  900      4-  600 
7000    4-  8000    4-  9000    4-  6000 
70000  4-  80000  4-  90000  +  60000 

etc. 

b.  25   4-  9   +  3   +  8 
250  4-  90  4-  3°  +  80 
2500  4-  900  4-  300  +  800 

etc. 

c.  25   +36   4-47   +  58 
250  4-  360  4-  470  +  580 
2500  4-  3600  4-  4700  4-  5800 

etc. 

d.  254   +  6   4-  8   4-  9 
2540  4-60  4-80  4-90 
25400  4-  600  4-  800  4-  900 

etc. 

e.  254  4-27  4-38  +49 
2540  4-  270  4-  380  4-  490 
25400  +  2700  4-  3800  4-  4900 

etc. 

/.     254   4-  316   +  449 
2540  4-3160  4-  449° 
25400  4-  31600  4-  44900 
etc. 


SECOND  HALF  OF  THE  YEAK. 


129 


The  pupils  must  be  led  step  by 

step  in  written  addition 

as  well  as  oral. 

Let  them  write  in 

columns  : 

(i)                      36 

Shorter  :     36 

24 

24 

15 

15 

23 

23 

50 

50 

18 

units 

148 

13 

tens 

148 

units 

(2)                            365 

or      365 

Shorter  :    365 

21 

21 

21 

1430 

1430 

H30 

2045 

2045 

2045 

320 

320 

320 

3000 

II 

4181 

1000 

170 

170 

1000 

II 

3000 

4181 

4181 

(3)                  5946 

5946 

5946 

847 

847 

847 

239 

239 

239 

6320 

6320 

6320 

2200 

130 

PX*)W 

13352 

130 

22OO 

22 

1  1  000 

13352        13352 

(Place  the  number  to  be  "carried"  under  the  column 
to  which  it  belongs,  in  parenthesis,  using  smaller  figure.) 

Pupils  should  be  able  to  answer  promptly  questions  on 
the  addition  processes. 


I3O  THE    THIRD    YEAR:    100-1000. 

THIRD  STEP. 
Subtraction. 

ORAL  AND  WRITTEN. 

a.  Numbers  of  one  figure : 

9  units  —  5  units  =  4  units  (9  —  5  =  4). 
9  tens  —  5  tens  =  4  tens  (90  —  50  =  40). 
9  hund.  —  5  hund.  =  4  hund.  (900  —  500  =  400). 
etc. 

b.  Numbers  of  two  figures  with  one  figure  : 

12  units  —  5  units  =  7  units  (12  —  5  =  7). 
12  tens  —  5  tens  =  7  tens  (120  =  50  =  70). 
12  hund.  —  5  hund.  =  7  hund.  (1200  —  500  =  700). 
etc. 

Continue  in  the  same  manner  as  in  addition. 
From  9456  take  7321. 

a.    9456  =  9000  +  400  +  50  +  6  =  9456 
7321  =  7000  +  300  +  20  +  i  =  7321 


2000  +  100  -f  30  +  5  =  2135 

b.    4325  —  1123  ==  4325  Minuend. 

1123  Subtrahend. 

3202  Remainder. 
c.  (Borrowing  without  ciphers.) 

16  15   18 

(I)  17-6-8*  1768 

679  679 


1089  1089 

(2)  4  5-4-2  45'4'2 

4159  4159 


383  383 


*  Grube"  placed  dots  at  the  bottom  where  "  borrowing"  was  employed  ; 
but  in  order  to  avoid  confusing  them  with  the  later  use  of  the  decimal 
point,  we  place  them  at  the  top.  The  Germans  use  the  comma  for  the  deci- 
mal point,  therefore  this  danger  does  not  arise  with  them. 


SECOND  HALF  OF  THE   YEAR.  13! 

Where  "borrowing"  is  necessary,  make  the  necessary 
changes  in  the  minuend,  placing  smaller  figures  above  to 
indicate  the  changes.  Then  work  the  same  example 
without  making  the  changes. 

d.  (Borrowing  with  ciphers.) 

(1)  7  4- 6-0       74-6-0 
3269^  3269 

4191       4I91 

(9  units  cannot  be  taken  from  o  units,  so  I  borrow  from 
the  tens ;  that  is,  take  one  from  the  6  tens,  which  I  indicate 
by  a  point  after  the  6.  The  ten  taken  away  =  10  units. 
9  units  from  10  units  =  i  unit ;  etc.) 

10 16 

(2)  74*0-6   74-06 
3269   3269 

4137   4137 

10  13  16 

(3)  7'04-6   7-04-6 
3269   3269 


3777   3777 

9 
10 10 

(4)  74-00   74-00 

3269   3269 


4i 31       4i 31 

*  9  9 
6101014 

(5)  7-004   7-004 

3269   3269 


3735   3735 


132  THE    THIRD    YEAR:   100-1000. 


9  9 

10  10  10 


(6)  7*0  o  o  __  7-0  o  o 

3269-3269 


3731      3731 

999 

10  lo  10 16 

(7)  7*0  o  o  6  __  7*0  006 

32697-32697 

37309      37309 

This  kind  of  work  must  not  be  left  until  the  pupil  is 
able  to  perform  it  rapidly.  He  must  understand  that  as 
7  units  cannot  be  taken  from  6  units,  he  must  borrow  of 
the  tens.  As  there  are  no  tens,  hundreds,  or  thousands, 
he  must  go  to  the  tens  of  thousands  to  borrow,  and  reduc- 
ing one  of  this  to  the  next  lower  denomination,  then  bor- 
rowing from  that,  etc.,  until  we  come  to  the  order  where 
we  need  to  increase  the  minuend  figure. 

The  explanation  of  subtraction  is  based  on  addition. 
Take  two  numbers : 

a.  3480 

b.  2375     Adding. 

c.  5855 

How  large  is  a  ?    (a  =  5855  —  2375.) 

How  large  is  bt    \b  =  5855  —  3480.) 

How  was  c  found  ?     (By  adding  a  and  b.) 

How  were  a  or  b  found  ?    (By  subtracting  the  known  a 

or  b  from  their  sum  c.) 

What  do  we  call  c  in  this  subtraction  ?   (The  minuend.) 
What  do  we  call  the  a  or  b  ?     (The  subtrahend.) 
What  do  we  seek  ?     (The  remainder  or  difference.) 
If  the  minuend  were  unknown  how  would  you  find  it  ? 

(By  adding  the  subtrahend  and  difference.) 

5855  5855 

-  3480  -  2375 

\ 
Proof. 


SECOND  HALF  OF   THE    YEAR.  133 

How  can  I  prove  subtraction  ? 

What  is  the  whole  in  an  example  in  subtraction  called  ? 
(The  minuend.) 

In  addition?     (The  sum.) 

What  is  the  difference  between  addition  and  subtrac- 
tion ?  (In  addition  the  parts  are  given  from  which  the 
whole  is  to  be  found ;  in  subtraction  the  whole  and  one 
of  the  parts  are  given  from  which  the  other  part  is  to  be 
found.) 

The  sum  5855  is  composed  of  three  numbers ;  the  first 
=  1320,  the  second  =  1427  ;  what  is  the  third  ? 

5855  r  132°  -  H27  =  5855  -  (1320  +  1427.) 
5855  1320      Proof:    3108 

1320  +  1427  ,  \  1320 

*  \  1427 

4535        2747 
1427  5855 

5855 
3108      -  2747 

3108 

FOURTH  STEP. 
Multiplication. 

ORAL  AND  WRITTEN. 
1.  Multiplier  One  Figure. 

a.  Multiplicand  One  Figure. 

3x9  units  =  27  units  (3x9  —  27). 
3x9  tens  =  27  tens  (3  x  90  =  270). 
3x9  hund.  =  27  hund.  (3  x  900  =  2700). 
etc. 

b.  Multiplicand  Two  Figures. 

3  x  29  units  =  87  units  (3  x  29  =  3  x  20  =  60 

3  x    9  =  27 

87). 

3  x  29  tens  =  87  tens  (3  x  290  =  870). 
3  x  29  hund.  =  87  hund.  (3  x  2900  =  8700). 


134  THE    THIRD   YEAR:   100-1000. 

c.  Multiplicand  Three  Figures. 

3  x  529  units  =  1587  units  (3  x  500  =  1500 
3  x  20  =  60 
3  x  9  =  27 

1587). 

3  x  529  tens  =  1587  tens  (3  x  5290  =  15870). 
etc. 

d.  Multiplicand  Four  Figures. 

3  x  5293  units  =  15879  units  (3  x  5000  =  15000 

3  x  200  =  600 
3  x  90  =  270 
3  x  3=9 

15879). 

3  x  5293  tens  =  15879  tens  (3  x  52930  =  158790). 
3  x  5293  hund.  =  15879  hund.  (3  x  529300  =  1587900). 


2.  Multiplier  Two  Figures. 

a.    THE  PURE  TENS. 
i.  Multiplicand  One  Figure. 

60  x  5  units  =  300  units  (60  x  5  =  300). 

60  x  5  tens  =  300  tens  (60  x  50  =  3000). 

60  x  5  hund.  =  300  hund.  (60  x  500  =  30000). 

60  x  5  thous.  =  300  thous.  (60  x  5000  =  300000). 

2.  Multiplicand  Two  Figures. 

60  x  56  units  =  3360  units  (60  x  50  =  3000 

60  x    6  =    360 


60  x  56  tens  =  3360  tens  (60  x  560  =  33600). 
etc. 


SECOND  HALF  OF   THE   YEAR.  135 

3.  Multiplicand  Three  Figures. 

60  x  562  units  =  33720  units  (60  x  500  =  30000 

60  x    60  =    3600 

60  X         2  =        I2O 

33720). 

60  x  562  tens  =  33720  tens  (60  x  5620  =  337200). 


b.    MIXED  TENS. 

(1)  25  x  9  units  =  225  units  (20  x  9  =  180 

5x9=    45 

225).   - 
25  x  9  tens  =  225  tens  (25  x  90  =  2250). 

(2)  25  x  96  units  =  2400  units  (20  x  96  =  1920 

5  x  96  =  480 


2400). 
25  x  96  tens  =  2400  tens  (25  x  960  =  24000). 

3.  Multiplier  Three  Figures. 

a.    PURE  HUNDREDS. 
i.  Multiplicand  One  Figure. 

300  x  9  units  =  2700  units  (300  x  9  =  2700). 
300  x  9  tens  =  2700  tens  (300  x  90  —  27000). 
300  x  9  hund.  =  2700  hund.  (300  x  900  =  270000). 

2.  Multiplicand  Two  Figures. 

300  x  91  units  =  27300  units  (300  x  90  =  27000 

300  x    i  =      300 

27300 
300  x  91  tens  =  27300  tens  (300  x  910  =  273000). 


136  THE    THIRD   YEAR:   100-1000. 

3.  Multiplicand  Three  Figures. 

300  x  914  units  =  274200  units  (300  x  900  =  270000 

300  x    10  =     3000 
300  x      4  =      1 200 


274200). 
300  x  914  tens  =  274200  tens  (300  x  9140  =  2742000). 

b.    MIXED  HUNDREDS. 

304  x  9  units  =  2736  units  (300  x  9  =  2700 

4x9=     36 

2736) 

304  x  9  tens  =  2736  tens  (304  x  90  =  27360). 
304  x  9  hund.  =  2736  hund.  (304  x  900  =  273600). 

This  will  be  sufficient  to  illustrate  the  method.     Prac- 
tice must  be  given  in  rapid  reckoning — 

a.  In  connection  with  addition  and  subtraction. 

b.  In  multiplication  alone. 

Method  for  Slate  Work. 

i.  Multiplier  One  Figure. 
(i)  No  carrying: 

3  x  3213  =  3  x  3000  =  9000 
3  x  200  =  600 
3  x  10  =  30 
3  x  3  = 9 

9639 

Shorter :         Shortest : 
3213  3213 

3  3 

9  9639 

30 
600 
9000 

9639 


SECOND   HALF  OF    THE   \EAR. 

(2)  Carrying: 

3  x  3226  =  3  x  3000  =  9000 
3  x  200  =  600 
3  x  20  =  60 
3  x  6=  18 

9678 

3226  3226 

3  3 

1  8  9678 

60 
600 
9000 


(3)  Cipher  in  Multiplication  : 

3  x  4046  =  3  x  4000  =  12000 
3  x  40  =  1  20 
3X  6=  18 


12138 

4046  4046 

3  3 


18  12138 

120 

I2OOO 


I2I38 

3  x  32130  =  (3  x  3213)  x  10 

32I3 
3 

9639  10  x  9639  =  96390 

Therefore:    3  x  32130  =  96390. 

The  pupil  will  learn  that  every  time  a  cipher  is  added 
to  the  product,  it  is  multiplied  by  10.     Therefore  it  fol- 


13$  THE    THIRD   YEAR:    100-1000. 

Jows  that  when  the  multiplier  is  tens,  hundreds,  etc.,  they 
may  be  treated  as  units  and  as  many  ciphers  added  to  the 
result  as  the  order  requires. 
For  example : 

345  x  2068  =  2068 
345 


103  4  o 

827  2(0) 

620  4(0  o) 

713460 


Explanation  of  Multiplication. 
How  many  times  must  I  take  112  to  get  336? 

112 
112 
112 

336 

How  can  I  express  that  shorter? 
3  x  112  =  336. 

112  a. 
_Z_b. 

336  c. 

How  many  numbers  have  I  ? 
Which  number  is  the  entirety?  (c.) 
What  is  the  a  ?  (A  part  of  c.) 

What  does  the  b  tell  us  ?     (How  many  times  we  must 
take  the  part  a  in  order  to  get  *:.) 
What  is  c  ?     (The  product.) 
How  do  we  get  c  from  a  ? 


SECOND  HALF  OF   THE   YEAR.  139 

What  do  we  call  the  number  which  we  multiply  in  order 
to  get  the  product  ? 

What  is  the  multiplicand?     Multiplier?     Product? 

The  multiplicand  and  multiplier  are  together  called 
factors,  because  they  produce  the  product. 

What  are  the  factors  of  620,  1000  ? 

If  336  =  3  x  112,  how  many  times  must  I  be  able  to 
subtract  112  from  336? 

336 

—  112 

224 

—  112 

112 

—  112 

000 

The  pupil  must  be  able  to  explain  multiplication  some- 
what as  follows : 

209  x  3148  =  ? 

The  multiplicand  is  3148.  This  is  a  part  of  the  un- 
known product.  The  multiplier  is  209,  and  tells  me  how 
many  times  I  must  take  the  multiplicand  3148  in  order 
to  get  the  product.  I  take  therefore  3148  209  times.  I 
multiply  first  by  9  units  and  get  28,332.  I  proceed  to  the 
tens,  and  as  there  are  none,  pass  on  to  the  hundreds.  I 
multiply  3148  by  2,  and  this  product  by  100,  and  this  gives 
me  629600.  Add  the  products,  and  I  get  657,932,  the 
product  of  209  x  3148. 

3H8 
209 

28332 
6296 

657932 


140  THE    THIRD   YEAR:    100-1000. 

It  should  by  shown  that  the  factors  may  change  places 
without  changing  the  result. 

312  (a)  113  (a) 

113  0)  312  (ff) 

936  226 

312  113 

312  339 


35256  (c)  35256  (c) 

FIFTH  STEP. 

Division. 

ORAL  AND  WRITTEN. 
A.— Without  Remainder. 

I.  DIVISOR   ONE  FIGURE. 

(Quotient  unchanged.) 
a.  Dividend  also  One  Figure. 
3  units  in  6  units     =  2  (6  -5-  3  =  2). 
3  tens  in  6  tens       =  2  (60  •*-  30  =  2). 
3  hund.  in  6  hund.  =  2  (600  -*-  300  =  2). 
etc. 

b.  Dividend  Two  Figures. 

3  units  in  18  units     =  6  (18  -*-  3  =  6). 
3  tens  in  18  tens        =  6  (180  -*-  30  =  6). 
3  hund.  in  1 8  hund.  =  6  (1800  •*•  300  =a  6). 
etc. 

£.  Dividend  Three  Figures. 

3  units  in  186  units      =  62  (186  -*•  3  =  62). 
3  tens  in  186  tens        =  62  (1860  -j-  30  =  62). 
3  hund.  in  186  hund.  =  62  (18600  •*-  300  =  62). 
etc. 

(Quotient  increasing  according  to  decimal  scale.) 
a.  One  third  of  6  units    =  2  units   (6  -5-  3  =  2). 
One  third  of  6  tens     =  2  tens     (60  -*-  3  =  20). 
One  third  of  6  hund.  =  2  hund.  (600  -*-  3  =  200). 


SECOND  HALF  OF  THE  YEAR.      14! 

b.  One  third  of  18  units  =  6  units  (18  -7-3  =  6). 
One  third  of  18  tens   =  6  tens   (180  -f-  3  =  60). 

c.  One  third  of  186  units  =  62  units  (186  -*-  3  =  62). 
One  third  of  1 86  tens   =62  tens    (1860-7-3  =  620). 


2.    DIVISOR  TWO   FIGURES. 

(Quotient  the  same.) 
a.  Dividend  also  Two  Figures. 

1 8  units  in  54  units    =  3  (54  -7-18  =  3). 
18  tens  in  54  tens       =  3  (540  •*-  180  =  3). 
18  hund.  in  54  hund.  =  3  (5400  -*-  1800  =  3). 

b.  Dividend  Three  Figures. 

18  units  in  108  units  =  6  (108  -f-  18  =  6). 
18  tens  in  108  tens     =  6  (1080  -f- 180  =  6). 

(Quotient  increasing.) 

a*  xV  of  54  units    =  3  units    (54-7-18  =  3). 
TV  of  54  tens     =  3  tens      (540  -f- 18  =  30). 
TV  of  54  hund.  =  3  hund.    (5400  •*- 18  =  300). 

b.    TV  of  108  units  =  6  units  (108  -7-18  =  6). 
•^  of  108  tens  =  6  tens    (1080  -5- 18  =  60). 
etc. 


3.   DIVISOR  THREE  FIGURES. 

(Quotient  the  same.) 
a.  Dividend  also  Three  Figures. 

114  units  in  342  units  =  3  (342  -r-  114  =  3). 
1 14  tens  in  342  tens     =  3  (3420  -T-  1 140  =  3). 
etc. 

b.  Dividend  Four  Figures. 

506  units  in  1012  units  =  2  (1012  -s-  506  =  2). 
506  tens  in  1012  tens    =  2  (ioi2o-r-  5060=  2). 


142  THE    THIRD   YEAR:  100-1000. 

(Quotient  increasing.) 

a.  Ti4  of  342  units  =  3  units  (342  -*-  114  =  3). 
•rff  of  342    tens  =  3  tens    (3420  -5-114  =  3o)« 


etc. 

I012  units  =  2  units  (1012  -j-  506  ==  2). 
1012  tens  =  2  tens    (10120  -*•  506  =  20). 
etc. 


B.-With  Remainder. 

I.    DIVISOR  ONE  FIGURE. 
(Quotient  also  one  figure.) 
a.  Dividend  One  Figure. 
3  units,  in  7  units  =  2  with   i  unit  remainder  (7  -5-  3  = 

2  [I]). 

3  tens  in  7  tens  =  2  with  i  ten  remainder  (70  -5-  30  = 
2  [10]). 

3  hund.  in  7  hund.  =  2  with  i  hund.  remainder  (700  •*• 
300  =  2  [100]). 

b.  Dividend  Two  Figures. 

3  units  in  25  units  =  8  with  i  unit  remainder  (25  •*•  3  = 
8  [i]). 

3  tens  in  25  tens  =  8  with  i  ten  remainder  (250  -f-  30  = 
8  [10]). 

etc. 

The  teacher  will  easily  continue  this  work  according  to 
the  plan  followed  in  the  preceding  pages  under  A. 

Rapid  reckoning  must  not  be  neglected. 

T\-  of  48  is  what  part  of  120? 

Y^y  x  1018  divided  by  2  is  what  part  of  100? 

|  x  9600  -*-  2  is  contained  how  many  times  in  9600  ? 

8000  -5-  800  x  3  x  (i  x  1 6)  is  how  many  times  12? 

3  x  120  -?-6x  5-5-15  x4-*-i6? 

55-5-18,  the  remainder  9  times  is  contained  how  many 
times  in  5409? 

etc. 


SECOND  HALF  OF  THE   YEAR.  143 

METHOD   FOR  WRITTEN   DIVISION. 
A.-Without  Remainder. 

I.    DIVISOR   ONE   FIGURE. 

a.  Dividend  Without  Ciphers. 
J5936  -5-3  =  3  in^o  15000  +  900  +  30  +  6. 

Shorter : 

3  into  15000  =  5000       3  into  15936  =  5000 

3  "   900=  300            15000  300 

3  "    30  =   10  10 

3  "     6  =   2             936  2 

900      

3    "    15936  =  5312                              —  5312 

36 
30 

6 
6 

Shorter : 


Ltoi5936  =  5-- 
15..-       3-. 

T 

or, 

3)15936 

3  into  15936(5312 
15... 

9..              2 
9..       
5312 
3- 
3- 

6 

The  shortest  : 

9.. 
9-. 

3- 
3- 

6 
6 

the  whole. 

the  part. 

5312 
15936 

Or  in  fractional  form  : 


144                 THE    THIRD  YEAR:   100-1000. 

6  into  49686(8  ...  or,          6  into  49686(8281 

48000    2  . .  48  ... 

8.  

1686        i 

1200 

8281  

486  48 . 

480  48 . 

~6  ~6 

6  6 

49686 
In  fractional  form  :   =  8281 


b.  Dividend  with  Ciphers. 

8  into  650048(80000 
640000  looo 

200 

10048   50 
8000    6 


2048  81256 
1600 

448 
400 


48 


(i  of  65  ten  thous.  =  8  ten  thous.  8x8  ten  thous.  = 
64  ten  thous.  64  ten  thous.  from  65  ten  thous.  leaves  i 
ten  thous.  £  of  10  thous.  =  i  thous.,  etc.) 


SECOND  HALF  OF   THE   YEAR. 

8)650048(81256         8)664800(83100 
64  ....  64  ....  


8.. 
8.. 

oo 


44- 
40. 

"48 
48 


c.  Divisor  with  Ciphers. 
10  into  664800  =  10  into  600,000  +  60,000  -f  4000  -f  800. 

10  into  60000  o  =  60000 
10    "      6000  o  =   6000 
10 
10 


4000  = 
800  = 


400 
80 


10  into  66480  o  =  66480 

Every  time  a  cipher  is  cut  off,  the  number  is  divided  by 
10,  for  it  makes  the  number  just  so  many  orders  lower. 

8^)6648^(8310 
64... 


24.. 
24.. 

8. 
8. 

o 
o 


100)664800  =  6648 


146 


THE    THIRD   YEAR.    100-1000. 


8^)6648^(83 1 
64.. 


24. 
24. 

8 
8 


664*$ 


=  831 


B.-With  Remainu^r. 

(Same  plan  as  with  A.) 
a.     59634  -f-  7. 

59634  =  59000  +  600  +  30+4. 

7  into  59000  =  8000 
56000 


3000 
+  600 

7)3600=  .500 
3500 

100 

+  30 


7)130  = 
70 

60 

+  4 

7)64  = 
63 

7)  i  = 


10 

9 
i 


=  8519^ 


SECOND  HALF  OF   THE   YEAR. 
7)59634(8519} 


147 


36.. 
35-. 

13.  or 

7- 

64 
63 


59634 


=  8519! 


b.  9  into  735040(81671$ 
72.... 


64. 


IO 

9 


735040 


•  =  81671$ 


c.  90  into  735040(8167$ 
72... 


15.. 
9" 

60. 
54 


735040   73504 

= =  8167$ 

90      9 


9^)73504^=73504 
64  =  8167$ 

63  9 


148  THE    THIRD  YEAR:   100-1000. 

90)73504(8161 
72000  •  •  •  ' 


1504 
900)735040  =  73504  900 


90  604 

540 

tt 

In  the  same  manner  the  teacher  will  be  able  to  continue 
these  operations,  completing  all  the  steps  in  division. 
Care  must  be  taken  that  the  pupil  be  always  clear  and 
prompt,  when  called  upon  to  explain  his  operations. 

EXPLANATION  OF  DIVISION. 

The  teacher  in  explaining  division  must  start  with  mul- 
tiplication. 

1260      (a) 
3    x  (b) 

3780       (c) 

What  is  c?    (The  product  or  whole.) 
What  is  a  ?    (The  multiplicand,  a  part  of  the  whole.) 
What  is  bl     (The  multiplier,  which  tells  me  how  many 
times  the  multiplicand  is  to  be  taken.) 
Find  a  from  b  and  c. 

(b)  (c)   (a) 
3)3780(1260 
3-.. 

6.'.' 

18. 
18. 


SECOND  HALF  OF   THE   YEAR.  149 

The  product  or  entirety  is  the  dividend ;  the  number 
that  I  divide  by  is  the  divisor,  and  the  number  sought,  or 
result,  is  the  quotient. 

c  =  dividend, 

b  =  divisor, 

a  =  quotient. 

What  are  a  and  b  with  reference  to  the  product  or 
dividend  ?  (Factors.) 

What  are  given  in  division? 

(The  product  and  one  factor.) 

What  must  be  found  ?   (The  other  factor.) 

How  is  that  done  ?     (By  division.) 

If  a  were  the  known  factor,  how  shall  b  be  found? 

1260)3780(3 
3780 


How  many  times  can  I  take  1260  from  3780? 


What  number  is  found  3  times  in  3780? 

The  pupil  must  be  able  to  give  an  explanation  similar 
to  the  following : 

The  number  which  is  contained  3  times  in  3780  must 
be  \  of  3780.  I  find  -J-  of  3780  by  dividing  it  by  3.  The 
divisor  3  is  the  known  factor ;  the  dividend  3780,  the  prod- 
uct and  the  quotient  is  the  unknown  factor,  which  will 
be  found  when  I  divide  the  product  by  the  known  factor. 

3780  =  37  hund.  and  8  tens ;  £  of  37  hund.  =  12  hund. 
with  remainder  of  i  hund.;  i  hund.  +  8  tens  =  18  tens; 
\  of  1 8  tens  =  6  tens. 

Therefore  \  of  3780  =12  hund.  +  6  tens  =  1260.  The 
quotient  1260  is  the  number  which  is  contained  3  times  in 
3780.  Therefore  1260  can  be  taken  3  times  from  3780. 


CONCRETE  NUMBERS. 

We  now  need  no  especial  explanation  of  the  application 
to  concrete  numbers  either  for  the  teacher  or  for  the  pu- 
pils. It  is  recommended  that  there  shall  always  be  oral 
exercises  first,  before  the  blackboard  or  slate  be  used,  un- 
til the  pupils  are  familiar  with  the  expressions. 

I.    ADDITION. 

How  many  times  does  the  clock  strike  in  24  hours  ? 

a.  What  is  given  in  this  example? 

(24  hours,  the  time  in  which  the  clock  strikes.) 
What  do  you  know  about  the  striking  of  the  clock  ? 
(The  clock  strikes  i  at  one  o'clock,  2  at  two  o'clock,  3 

at  three  o'clock,  etc.,  till  12,  and  then  it  begins  to  strike 

i,  2,  3,  etc.,  again  to  12.) 

b.  What  is  required  ? 

(The  number  of  strokes  of  the  clock  in  24  hours.) 
As  the  clock  strikes  only  12,  how  shall  the  reckoning  be 
done  ? 

(Find  the  number  of  strokes  for  12  hours.) 
Then  how  do  you  find  the  number  for  24  hours  ? 
(By  taking  the  number  for  12  hours  twice.) 

c.  How  many  strokes  for  12  hours? 

(1  +  2  +  3  +  4  +  5  +  6  +  7  +  8  +  9  +  10  +  11  +  12  = 
78  strokes.) 

How  many  for  24  hours  ?     (2  x  78  =  156  strokes.) 

How  does  the  number  of  strokes  for  24  hours  compare 
with  those  for  12  hours? 

How  does  the  number  of  strokes  for  the  first  1 1  hours 
compare  with  those  of  the  12  hours  ?  Of  the  I2th  hour? 

How  many  strokes  have  there  been  at  the  1 5th  hour  ? 

(78  +  i  +  2  +  3  =  84  strokes.) 

Give  a  variety  of  concrete  examples. 

2.    SUBTRACTION. 

The  property  of  a  man  before  a  fire  consisted  of  34580 
dollars.  After  it  he  had  only  6594  dollars. 

How  much  did  he  lose  ? 

a.  How  do  you  see  already  that  the  man  has  lost  by  the 
fire? 


SECOND  HALF  OF  THE  YEAR.      !$! 

(He  had  more  money  before  the  fire  than  after  it.) 
How  much  must  he  have  had  after  the  fire  in  order  to 
be  able  to  say  that  he  had  lost  nothing  ? 
(He  must  have  had  34580  dollars.) 
But  how  much  had  he  ?     (He  had  only  6594  dollars.) 

b.  What  would  you  call  his  loss  if  he  counted  his  money 
and  found  only  $6594?  * 

(What  this  lacks  of  $34580.) 
Concerning  what  is  the  question  ?     (The  loss.) 
How  much  must  the  man  add  to  $6594  in  order  to  get 
$34580  ? 

c.  How  do  you  find  that? 

(By  adding  to  $6594,  until  I  get  $34580;  or  by  subtract- 
ing 6594  from  34580.) 

Do  the  first. 

6594  =  65  hund.  +  94  units;  I  make  the  66  hundreds 
complete  by  adding  6  units.  As  34580  =  345  hund.  +  80 
units,  I  must  have  with  the  66  hund.  34  hund.  +245  hund. 
+  80  units  more.  34  hund.  -f  245  hund.  =  279  hund.;  to 
the  80  units  must  be  added  also  6  units  =  86  units.  So 
I  must  add  to  6594  279  hund.  +  86  units  =  27986  in  order 
to  get  34580.  Therefore  the  man  must  have  lost  27986 
dollars. 

How  can  you  express  the  loss  by  using  the  numbers 
given  in  the  example  ? 

(The  man  had  lost  34580  —  6594  dollars.) 

The  loss  equalled  what  difference  ? 

(The  difference  between  his  first  and  last  amount  of 
property.) 

Show  in  the  same  way  how  much  greater  his  loss  was 
than  what  he  still  retained. 

Suppose  the  loss  to  be  known,  and  the  later  property 
unknown  ;  what  would  the  example  then  be? 

(The  property  of  a  man  was  $34580.  By  a  fire  he  lost 
$27986.  How  much  had  he  still  ?) 

In  the  same  way  suppose  the  property  at  first  to  be  un- 
known. How  would  the  example  read  ? 


*  The  use  of  the  dollar  sign  can  be  taught  here,  if  not  earlier.    Make  the 
sign  ($),  and  tell  the  children  that  it  stands  for  dollars. 


I$2  THE    THIRD   YEAR:   100-1000. 

3.   MULTIPLICATION. 

A  merchant  bought  3900  cwt.  of  wares  @,  $36,  and  sold 
them  again  @.  $42  per  cwt.  How  much  did  he  make  out 
of  the  transaction  ? 

a.  What  did  the  merchant  do  ? 
(He  bought  3900  cwt.  @  $36.) 

What  else  did  he  do  ?     (He  sold  i  cwt.  @  $42.) 
What  is  the  cost  price  and  the  selling  price  of  i  cwt.  ? 

b.  What  do  you  find  when  the  price  of  both  are  com- 
pared ? 

As  the  selling  price  is  the  greater,  what  do  we  find  ? 

(A  gain.) 

How  much  is  gained  on  i  cwt.  ?     (6  dollars.) 

How  do  you  find  that  ? 

What  is  required  in  the  example? 

(The  gain  of  the  whole  transaction.) 

That  is  how  many  cwt.  ?     (3900.) 

What  do  you  know  of  the  gain  ? 

(That  6  dollars  have  been  gained  on  i  cwt.) 

How  do  you  find  the  entire  gain  ?     (3900  x  6  dollars.) 

3900  x  6  =  6  x  3900  =  6  x  39  hund.  =  6  x  30  hund.  -f 
6  x  9  hund.  =  180  hund.  -f  54  hund.  =  234  hund.  =  23400. 
Therefore  $23400  is  the  entire  gain. 

1.  What  was  the  entire  cost ?    Selling  price ?    Gain? 

2.  The  gain  on  i  cwt.  was  $6.     What  was  the  selling 
price,  the  cost  being  $36?    The  selling  price  was  $42,  the 
gain    $6.     What   was    the   cost?     The   entire  gain   was 
$23400,  and  the  gain  on  i  cwt.  was  $6.     How  many  cwt. 
were  there  ? 

3.  If  the  merchant  had  gained  only  $11700,  what  would 
have  been  the  selling  price  per  cwt.  ? 

(If  he  gains  $11700  on  3900  cwt.,  on  i  cwt.  he  gains  ^-gVfr 
of  $11700  =  $3.  Since  i  cwt.  cost  $36  and  gain  $3,  he 
must  have  sold  for  $36  4-  $3  =  $39  per  cwt.) 

4.  DIVISION. 

A  gardener  worked  a  week  in  a  garden  and  received  for 
his  work  9  dollars  and  60  cents.  How  much  did  he  re- 
ceive daily  ? 

a.  There  are  6  working  days,  for  which  he  receives  9 
dollars  and  60  cents. 


SECOND  HALF  OF   THE   YEAR.  153 

b.  We  must  find  the  reward  of  one  day's  labor. 

c.  If  in  6  days  he  earned  9  dollars  and  60  cents,  in   i 
day  he  will  earn  \  of  that.    \  of  9  dollars  is  i  dollar,  and 
3  dollars  remainder.     3  dollars  =  30  dimes  ;  \  of  30  dimes 
=  5  dimes ;  \  of  60  cents  =  10  cents.     Therefore  he  re- 
ceives i  dollar  +  5  dimes,  or  50  cents,  +  10  cents  =  i  dol- 
lar 60  cents  per  day. 

If  a  workman  receives  $1.60  for  a  day,  how  much  does 
he  receive  per  week  ? 

A  man  earns  $9.60,  earning  $1.60  per  day.  How  long 
does  he  work  ? 

How  does  the  pay  for  a  week  compare  with  that  of  a 
day  ? 

5.    MIXED   EXERCISES. 

i.  Two  merchants  compare  their  gain  after  a  transac- 
tion. B  said  to  A,  "  The  half  of  your  gain  is  one  third  of 
mine."  A  had  gained  $605.  How  much  had  B  gained  > 

a.  What  do  you  know  of  A's  gain  ?     (It  is  $605.) 
What  do  you  know  of  B's  gain  ? 

(That  t  of  it  =  J  of  A's.) 

b.  If  you  knew  J-  of  B's   gain,  what  could  you  easily 
find  ?     (His  whole  gain.) 

How  much  would  it  be?     (3  x  the  one  third.) 
But  we  have  been  told  of  what  amount,  which  is  equal 
to  \  of  B's  ?     (^  of  A's  gain.) 

How  much  is  that  ?     (^4p-  dollars  =  $302.50.) 

c.  How  much  is  B's  gain?    (3  x  $302.50  =  $907.50.) 

a.  B  gained  $907.50;  A.  said,  "-J-of  your  gain  =4  of 
mine."     What  was  A's  gain  ? 

b.  Two  merchants  compared  gains  and  found  that  B's 
gain  was  \  greater  than  A's.     A  had  gained  $605.     How 
much  B? 

c.  Two  merchants  compared  gains,  and  A  found  that 
his  gain  was  \  less  than  B's,  whose  gain  was  $907.50. 

d.  Two  merchants  in  comparing  gains  found  that  what 
A  had  gained  twice  B  had  gained  3  times.     B  had  made 
$907.50. 

e.  They  also  found  that  f  of  B's  gain  =  the  whole  of 
A's.     B  had  made  $907.50.     How  much  A  ? 


154  THE    THIRD   YEAR:    100-1000. 

2.  Three  persons  divide  4  cwt.  and  40  Ibs.,  so  that  A  re- 
ceives 30  Ibs.  and  B  20  Ibs.  more  than  C.     How  many 
pounds  did  C  get? 

(We  subtract  first  what  A  and  B  receive  extra,  that  is, 
30  Ibs.  +  20  Ibs.  =  50  Ibs.  440  Ibs.  —  50  Ibs.  =  390  Ibs. 
\  of  390  Ibs.  =  130  Ibs.,  C's  part.  B  receives  20  Ibs.  extra, 
making  150  Ibs.  A  receives  30  Ibs.  extra,  making  160  Ibs.) 

a.  Three  persons  divided  a  quantity  of  corn  so  that  A 
had  5  Ibs.  more  than  B,  and  B  10  Ibs.  more  than  C.     B's 
part  was  155  Ibs.     How  much  was  A's?     C's?    What  was 
the  whole  amount  divided  ? 

b.  Three  persons  divided  440  Ibs.,  so  that  A  received  10 
Ibs.  more  than  B,  and  C  20  Ibs.  less.     What  was  the  part 
of  each  ? 

c.  12  workmen  work  on  a  building,  4  carpenters  and  8 
masons.     The  carpenters  receive  each  40  cents  a  day 
more  than  the  masons.     The  pay  of  all  the  workmen 
amounts  to  25  dollars  60  cents  a  day.     How  much  does 
each  carpenter  and  each  mason  receive  per  day? 

(4  x  4oc.  =  $1.60.  $25.60  —  $1.60  =  $24.  $24  -T-  12  = 
$2,  what  each  mason  receives.  $2+4oc.  =  $2.40,  the  pay  of 
each  carpenter.) 

3.  N  bought  cloth  for  a  new  coat,  paying  $3  a  yard. 
The  whole  cost  was  $12  ;  how  many  yards  did  he  buy? 

a.  N  bought  cloth  for  a  coat,  paying  $3  a  yard.     If  he 
took  4  yards,  what  was  the  cost  ? 

b.  If  he  paid  $12  for  4  yards,  what  was  the  cost  per 
yard  ? 

c.  B  took  of  the  same  kind  of  cloth  4^  yards.     How 
much  must  he  pay  ? 

(If  he  had  taken  4  yards,  the  cost  would  be  $12.  J  yard 
costs  J  x  $3  =  75  cents.  $12  +  .75  =  $12.75,  the  amount 
he  must  pay.) 

The  teacher  should  multiply  examples  embracing  all  of 
compound  numbers,  until  the  pupil  is  prompt  and  accu- 
rate in  the  work.  Great  attention  must  be  paid  to  the 
solution  of  examples.  One  example  solved  understand- 
ingly  is  of  more  worth  than  a  dozen  solved  mechanically, 
according  to  rule,  without  understanding  the  principles 
involved. 


THIRD   COURSE 

FRACTIONS. 


THE  FOURTH  YEAR. 


FIRST  HALF  OF  THE  YEAR. 

GENERAL  CONTEMPLATION  OF  THE  FRACTION. 

REMARKS. 

1.  As  the  pupil  arrived  at  a  perception  of  whole  num- 
bers by  measuring  them  by  the  smallest  unit,  so  will  he 
come  to  comprehend  fractions  by  constant  reference  to  the 
number  one  from  which  they  have  arisen. 

2.  While  heretofore  the  one  has  been  considered  as  a 
part  of  other  numbers,  it  will  now  be  considered  as  a 
whole  consisting  of  parts.    These  parts  can  be  resolved 
into  their  elements.     With  reference  to  their  whole  they 
are  called  fractions. 

3.  As  the  pupil  has  learned  from  the  first  to  consider 
whole  numbers  as  fractions,  in  that  he  recognized  them  as 
parts  of  larger  numbers,  the  following  treatment  of  the 
real  fraction  (the  broken  unit)  will  offer  no  difficulty  to 
him.     The  process  is  exactly  the  same  as  that  he  has  used 
in  whole  numbers,  namely,  perception  of  the  manifold 
relations  in  their  organic  unity. 

4.  As  the  different    kinds  of    fractions  depend   upon 
their  size,  and  their  size  upon  the  number  of  equal  parts 


156  THE  FOURTH    YEAR. 

into  which  the  unit  is  divided,  the  different  kinds  of 
divisions  may  be  considered  as  especial  orders,  namely, 
descending  lower  orders,  as  in  whole  numbers  the  ascend- 
ing higher  orders  of  the  units,  tens,  hundreds,  etc.,  are 
formed  by  taking  unity  ten  times. 

5.  We  treat  in  the  first  step  the  half,  in  the  second  step 
the  third,  etc.,  until  the  pupil  through  this  natural  de- 
velopment of  his  perception  comes  to  the  observation  of 
the  fraction. 

6.  We  begin,  as  in  the  preceding  courses,  with  general 
observation  of  the  object,  and  practise  in  the  same  man- 
ner oral  and  written,  pure  and    applied,  blending  addi- 
tion, subtraction,  etc.,  together,  and  treat  the  fraction 
analogous  with  the  whole  number  under  the  following 
heads : 

1.  Contemplation  of  the  pure  number. 

a.  Measuring. 

b.  Comparing. 

c.  Combining. 

2.  Application  of  the  pure  relation  of  number  to  all  the 
fundamental  rules. 

FIRST  STEP. 

Halves. 

1. 

I I I  * 


*  (The  line  divided  into  parts  is  to  be  the  standard  illustration  for  frac- 
tions, though  other  things  may  also  be  used.  Avoid  withdrawing  the  at- 
tention by  attractive  objects,  remembering  that  all  of  the  attention  given 
to  the  object  is  so  much  withdrawn  from  the  subject  in  hand,  namely,  frac- 
tions. Give  many  practical  examples,  as  the  four  processes  are  carried 
along  together  from  the  first. 

In  division  of  fractions,  do  not  allow  the  pupil  to  speak  of  2  divided  by 
1^3,  4  by  %,  etc.,  as  he  cannot  understand  it  at  this  period  ;  but  rather,  how 
many  times  is  *&  contained  in  2,  or  %  contained  in  4  ?  The  child  can  be 
shown  how  many  times  ^  is  contained  in  2.  It  is  better  that  the  pupil  read 
4  H-  %:  4  is  twice  the  third  part  of  what  number;  or  better,  %j  are  con- 
tained ir  4  how  many  times  ?) 


FIRST  HALF  OF  THE  YEAR. 


157 


1                            1 

A                                                                  A 

^                               ^ 

A 

i 

i 

i 

1 

If  I  divide  one  (a  whole)  in  two  equal  parts,  I  get  2 
halves.  One  half  is  one  of  the  2  equal  parts  into  which 
the  whole  is  divided. 

i  -«-  2  =  i  or  i  x  i  =  i. 

MEASURING. 

a.  (Adding)  i  +  i  =  i. 

b.  (Multiplying)  i  x  -J-  =  £,  2  x  i  =  i. 

c.  (Subtracting)  i  —  i  =  i. 

d.  (Dividing)  i  -H  i  =  i,  1-7-^  =  2  (£  is  contained  in  i 
twice). 

APPLICATION. 

a.  Since  i  •*•  2  =  -J,  2  -i-  2  =  f ,  2  -i-  3  =  f ,  10  -f-  2  =  ^ , 
loo  -*-  2  =  i£4,  etc. 

ADDITION. 

1  +  i  =  i,  i  +  i  =  ii,  2  +  i  =  2i,  3  +  \  =  3i>  etc.;  i-J 
+  ^  =  2,  2^  +  i  =  3,  I2i  +  i  =  13,  etc.;  ii  +  ii  =  3  (for 
i|  +  i  =  2,  +  i  =  3,  or  i  +  i  =  2,  i  -f  |  =  i,  2  +  1=3), 
Sir  +  i*  =  7i   7i  +  8  =  isi,  7i  +  8i  =  16,  S  +  8i  =  i6i, 
etc. 

MULTIPLICATION. 

2  x  -J  =  f  =  1,3  x  -J  =  f  =  il,  10  x  i  =  -1/  =  5,  ioo  x 
£  =  loo  _  50;  7  x  i  =  |  =  3i,  73  x  -i-  =  ¥  =  361,  etc. 

i£  =  3  x  |  =  |  =  4i,   etc.    (or  3  x  i  =  3,  3  x  i  =  i|,  3 
x  ii  =  41). 

6xi5l  =  6xi5  +  6x£,  etc. 

9  x  8o£  =  9  x  80  +  9  x  i. 

As  |  x  i  =  i,  i  x  6  =  |  =  3,  i  x  9  =  |  s=  4J. 


158  THE  FOURTH  YEAR. 

SUBTRACTION. 

i  —  i  =  ii  2  —  i  =  ii  (for  2  —  i  =  i;i—  i  =  i,  i+i 
=  Ti)>  3  —  4  =  24,  2  —  14  =  4  (for  2  —  i  =  i  ;  I  —  4  =  4, 
i  +  4  =  1  4),  6  -  44  =  ii  (for  6  -  4  =  2,  2  -  4  =  ii),  9  - 
34  =  si,  etc.  2i  -  i  =  ii,  6i  -  3  =  3*  (  =  [6  -  3l  +  i), 
etc.  34  —  24  =  i  (3  —  2=1,  4  —  4  =  o  ;  or  34  —  2  =  14, 
ii-4=i).  84-4i=? 

DIVISION. 

•J  in  i  =  2  (for  i  =  f  and  i  in  f  =  2),  i  in  4  =  8  (for  4 
=  f  and  i  in  f  =  8  ;  or  i  in  i  =  2  and  4  in  4  =  4  x  2,  or 
8.)* 

i4  -4  =  1-4  =  3-1  =  3-  1 
94  -f-  i  =  V-  —  f  »  etc. 


COMPARING. 

J6  with  1. 
4=1-4,  1=4  +  4- 

i  =  the  half  of  i,  i  =  two  times  \. 

What  number  shows  me  the  difference  between  £ 
and  i  ? 

How  much  must  I  take  from  16  to  get  9!  ? 

One  of  two  numbers  is  9^  ;  the  difference  between  it 
and  a  greater  number  is  6J-;  what  is  the  greater  number? 

Give  two  other  numbers  whose  difference  is  6J,  4i,  94. 

How  many  times  must  I  take  \  to  get  i  ?  How  many 
times  44  to  get  9  ? 

Of  what  number  is  44  the  half  ? 

Of  what  number  is  9  the  double  ? 

The  divisor  is  44,  the  quotient  2,  what  is  the  dividend  ? 

What  number  must  I  take  4  times  to  get  44  ? 

*  Though  we  indicate  the  division  by  the  American  method,  the  expres- 
sion must  not  be  read  "  1^  divided  by  ^,"  but  "  J^  in  1J4,"  or  "/^z  con~ 
tained  in  1^£."  See  page  156. 

t  We  simply  state  the  fact  here  and  the  reason  therefor,  leaving  the 
teacher  to  choose  the  method  of  questioning.  The  method  employed  in 
the  whole  number  can  be  applied  here  very  well.  For  example  : 

How  many  halves  in  1  ? 

How  many  halves  in  3  ? 

How  many  halves  in 


FIRST  HALF  OF  THE   YEAR.  1 59 

APPLIED   NUMBER. 

What  is  \  a  dollar  ?  (^  a  dollar  equals  one  of  two  equal 
parts  into  which  I  divide  a  dollar.) 

How  many  half  dollars  in  17  dimes?  (i  dollar  =  5 
dimes,  17  dimes  -f-  5  dimes  =  3  [2  dimes].) 

In  a  hotel  17!  +  13^  +  &J  pounds  of  meat  were  bought. 
How  many  "portions"  will  this  make,  allowing  -J  Ib.  to 
a  portion  ? 

SECOND  STEP. 

Thirds. 

1. 


*•;...*.-.> 

If  I  divide  i  into  3  equal  parts,  one  part  is  \. 
•J-  is  one  of  the  3  equal  parts  into  which  I  have  divided  i. 
f  are  2  of  the  3  equal  parts  into  which  I  have  divided  i. 
3  in  i  =  i  or  \  x  i  =  £. 

a.  £  +  *  =  »,  »  +  t  =  t=i. 

b.  i  x  J  =  *,  2  x  i  =  f,  3  x  i  =  f  =  i. 
,.  i-i  =  *,»-i  =  i. 

*/.  3  in  i,  or  i  -4-  3  =  i,  2  -^  3  =  f,  i  -5-  i  =  I. 

1 4-  3  =  i,  2  +  3=  f,  ip  +  3  =  ¥• 

ADDITION. 

2  +  \  =  2l,  8  +  4i  =  i2t,  Si  +  4i  =  9l.  J7*  +  17*  =  35» 
i7*  +  i7t  =  35*.  etc. 

MULTIPLICATION. 

i  xt  =  t,9xt  =  f  =  3,  14  x  t  =  V  =  4t.  etc. 

i  x  f  =  f,  9  x  t  =  ¥  =  6, 14  x  t  ==  Y  ==  gfc  10  x  »  = 

y  =  6f,  etc. 

3  x  ii  =  4  (3  x  i  +  3  x  i,  or  3  x  it  =  3  x  f  =  ¥  =  4), 
9  x  i£=  12,  etc. 

3  x  if  =  5,  5  x  if  =  8i. 

As  i  x  i  =  i,  £  x  2  =  f,  -Jx6  =  f  =  2,  ix7  =  £  =  2i, 
etc. 

fx  i=f,  f  X2  =  t=it,  fx9  =  ¥  =  6,  f  x  ii  =V 
=  7i,  etc. 


160  THE  FOURTH  YEAR. 


SUBTRACTION. 

i  -  i  =  •£,  2  -  i  =  if,  etc. 

1  —  f  =  t,  2  —  f  =  ii,  etc. 

2  —  ij-  =  f  ,  4  —  ii  =  2f,  etc. 
7t-4*=3*. 

7t  -  4t  =  2t  (7  -  4*)  +  i,  or  7*  -  4  -  t- 

DIVISION. 

i  in  i  =  3,  i  in  2  =  2  x  3  =  6,  i  in  3  =  3  x  3  =  9,  etc. 

i  in  14  =  42  (i  in  i  =  3,  14  x  3  =  42). 

f  in  i  =  |  (i  in  i  =  3,  f  in  i  =  one  half  of  3  =  f). 

6  -*-  1  =  9  (6  -  i  =  1  8,  6  •*•  f  =  V  =  9). 
4*  -*-  2|  =  2  (-V-  •*•  I  =  H  •*•  7  =  2). 

20  -T-  6f  =  &$•  -T-  $£•  =  60  -T-  20  =  3. 

COMPARING. 
3^  w/M  1. 

*=!-»,  I=i+t. 
^  =  ^  X   I,   I  =  3  X  *. 

%  wzVA  1. 
f  =  I    -  ^  I  =  I   X  f  +  t. 


1 

»* 

1    1   ' 

L__|     *    I    * 

*  , 

^ 

1           1       ^1 

^  -.  -^v.      _ 

1 

Thirds  and  halves  are  common  (have  a  common  de- 
nominator) in  sixths. 


^  =  |  x  i  (twice  the  third  part  of  |),  for  |  or  |  in  ^  or 

(  =  3  in  2)  f  times. 

i  =  f  x  i  (3  times  the  half  of  1),  for  i  -s-  i  =  f  -f-  1  =  3 


FIRST  HALF  OF   THE   YEAR.  l6l 

X  with  %. 

*  =  *,*  =  *• 

1  =  f  —  i,  i  =  I  +  I- 

|  into  i  =  i,  for  f  into  i  =  4  into  3  =  f . 
f  =  I  x  i,  for  i  into  f  =  3  into  4  =  f  * 

3  wzV/f  2. 

3  is  i  greater  than  2 ;  i  is  £  of  2 ;  therefore  3  is  $  greater 
than  2.  (Remark,  i  of  2  greater;  the  pupil  is  to  learn 
that  |  does  not  always  mean  a  part  of  a  unit, — it  may  be  a 
part  of  a  whole  number.) 

If  a  boy  has  a  string  3  yards  long,  it  is  i  longer  than  a 
string  2  yards  long. 

3  hundred  dollars  is  i  more  than  2  hundred  dollars,  3 
thousand  dollars  than  2  thousand  dollars,  etc. 

Three  is  therefore  one  half  greater  than  two. 

2  is  i  less  than  3.     i  is  \  of  3.     2  is  therefore  \  less  than 
3.     (Remark,  i  of  3  less  ;  see  remark  above.) 

If  Henry  has  a  string  2  yards  long  and  Peter  one  3 
yards  long,  Henry's  is  \  shorter  than  Peter's.  (See  remark 
above.) 

If  I  have  $200,  I  have  \  less  than  a  man  who  has  $300. 

Two  is  one  third  of  3  less  than  three. 

Give  further  examples. 


The  first  line  indicates  2x1=2,  the  second  3x1  =  3; 
a  £,the  part  of  3  greater  than  2.  The  part  ab  is  \  of  2  ; 
it  equals  also  i  of  3. 

That  is,  the  three  has  3  of  the  same  parts  of  which  the 
two  has  only  2. 

*  The  teacher  must  not  forget  that  this  work  is  chiefly  mental,  and  does 
not  need  the  use  of  slate.  The  above  diagram,  which  appeals  to  the  eye, 
will  remove  all  seeming  difficulties. 


i62  THE  FOURTH  YEAR. 

2  -with  3. 


a 

2 


The  two    =  2 
The  three  =  3 

What  is  the  relation  of  $3  to  $2  ? 

($>3=f-of$2,$2=f  of  $30 

What  part  of  a  yard  are  2  feet  ? 
2  feet.  I 1 1 


i  yard.  | 1 

2  feet  =  f  of  i  yard, 
i  yard  =  |  of  2  feet. 

How  many  times  is  |  —  £  contained  in  i  ?  2  ?  3  ? 

How  many  times  is  \  —  \  contained  in  f  ? 

How  many  times  must  I  take  -J  to  get  8  ? 

8  x  \  is  how  much  more  than  8  x  £  ?     How  much  less 
than  8  x  -f  ? 

How  much  is  f  of  100  Ibs.? 

How  many  pounds  more  in  $•  than  \  of  100  Ibs.? 

(i  of  100  Ibs.  =  331  Ibs.;  f  =  66f  Ibs.;  ^  of  100  Ibs.  =  50 
Ibs.     66f  Ibs.  —  50  Ibs.  =  i6f  Ibs.) 

How  many  packages  of  tea,  each  weighing  \  lb.,  can  be 
made  from  1 5  Ibs.  ?    £  lb.  packages  ? 

THIRD  STEP. 

Fourths. 

1. 


1  i 

If  I  divide  i  into  4  equal  parts,  one  part  =  }.    4  in  i  = 
1,  i  x  i  =  i. 


FSXST  HALF  OF  THE  YEAR. 

ADDITION. 

j.  =  1,  f  +  i  =  f ,  f  +  i  =  |=f  =  ] 
MULTIPLICATION. 

SUBTRACTION. 


DIVISION. 

J  in  i  =  i,  1  in  £  =  2,  J  in  f  =  3,  i  in  i  =  4. 
i  •*•  4  =  t,  2  -t-  4  =  f  =  i,  3  -*-4  =  t- 

MIXED. 

4i  +  f  =  4  +  i  =  4  +  i  =  5- 

4i  +  4i  =  8f  ,  etc. 

i  x  1  =  1,9  xi=|  =2i. 

9x  ii  =  9  +  1  =  9  +  2  +  1  =  TIt- 

9  x  3f  =  27  +  V-  =  27  +  6|=  33f- 

1  x  9  =  J,  f  x  9  =  ^,  f  x  16  =  -4/  =  12. 

i-i  =  f,  i6-i  =  i5£. 

20  —  f  =  I9J,  20f  —  i  =  20}. 

20  —  6f  =  13!,  2of  —  6i-  =  14!. 

i-!-i  =  4,  8-f-i=  8  x  4  =  32,  32  -*-J  =  4  x  32=  128. 

5  -^  1  =  ^a.  -f-  J  =  20  -f-  i  =20. 

5*  -*-  i  =  ¥  -*•  i  =  23  •*•  i  =  23- 
25^6^  =  4(25=^,61  =  ^). 

27  x  4-*- 


2,  3,  4,  5,  6  -i-  f  . 


TV  TV 


A  TV 


Fourths  and  thirds  have  a  common  denominator  in 
twelve. 


1  64  THE  FOURTH  YEAR. 


=  |  x  i,  f  or  i  -*-  i  =  t  (T%  - 


=  f  x  f  (i  of  I-  taken  3  times),  for  J  -5-  f  =  f  . 
=  f  x  i,  for  »  -*-  J  (ft  -  ft). 

%  with  %. 


f  =  i  x  f  for  *  -H  *  =  f  =  i*. 

f  =  f  x  i,  for  *  -5-  f  =  |  (T%  -^-  A). 

What  is  the  common  denominator  of  halves,  thirds,  and 
fourths  ? 

i  =  if,  i  x  if  =  TV  i  x  if  =  A,  i  x  Jf  =  A. 

What  relation  do  the  numbers  3  and  4  bear  to  each 
other? 

3  =  3x1,4  =  4x1.  i  =  J  x  4,  3  =  fx4.  i  =  *  x 
3,  4  =  f  x  3.  4  is  i  times  greater  than  3,  and  3  is  J  times 
smaller  than  4. 

Show  the  same  relation  with  3,  6,  9,  10  times  3  and  4. 

Two  numbers  have  a  sum  of  i6-^c  ;  one  is  6f.  What  is 
the  other  ? 

(i6T\  —  6£  =  16  -  6f  +  T\  .       16  —  6  =  10,  10  —  f  =  # 

+  T5*  =  9A  =  9*0 
The  difference  between  io£f  and  an  unknown  number 

is  9|. 
What  is  the  unknown  number  ? 

(16*  -  9*  =  16  -  9*  +  *0 

Suppose  i6^y  to  be  the  smaller  number,  and  the  differ- 
ence 9!  ;  what  is  the  larger  number  ? 


What  is  the  relation  of  i  cwt.  to  f  cwt.  in  whole 
numbers? 

How  many  times  must  I  take  f  cwt.  in  order  to  get  I 
cwt.  ? 


FIRST  HALF  OF   THE   YEAR.  165 

(As  the  relation  of  f  cwt.  to  i  cwt.  is  as  3  to  4,  I  must 
take  4  times  i,  or  f  of  f  cwt.) 

How  many  times  must  I  take  6£  to  get  8£? 

(As  often  as  6J  is  contained  in  8£.  6£  =  ^ ;  8&  =  \5  ; 
jyi  +  y  —  j -*.$=-$.) 

How  many  times  must  I  take  8£  to  get  6J? 

(As  many  times  as  8i  is  contained  in  6J,  etc.) 

\  of  6J  is  i  of  what  number  ? 

I  of  8£  is  what  part  of  6i  ? 

If  I  take  12  from  a  number  and  still  have  \  of  the  num- 
ber left,  what  is  the  number  ? 

(Since  I  still  have  i,  I  must  have  taken  f  away;  so  12 
=  f  of  the  unknown  number,  and  £  or  the  whole  equals  4 
times  \  of  12  or  16.) 

What  is  the  relation  of  the  12  to  the  whole  number? 

What  is  the  relation  of  the  remainder  to  the  subtra- 
hend ? 

How  many  pounds  in  f  cwt.  ? 

How  many  oz.  in  f  Ib.  ? 

How  many  pwt.  in  f  oz.  ? 

How  much  more  is  |  +  \  +  i  of  a  dollar  than  f  of 
a  dollar  ? 

N  had  $100  to  spend  in  travel.  How  long  can  he  travel 
if  he  spend  i£  dollars  per  day  ? 

(He  can  travel  as  many  days  as  i£  dollars  are  contained 
in  loo  dollars,  ij  =  £.  i  in  100  =  400 ;  f  =  £  of  $400  or 
$80.  Or  $  i  oo-s-f  =  §80.) 

N  was  2-f  months  on  a  journey  and  spent  $100.  How 
much  did  he  spend  per  day  ? 

N  was  2f  months  on  a  journey,  spending  ij  dollars  a 
day.  How  much  did  his  journey  cost  him  ? 

A  and  B  gave  a  poor  family  some  money.  A  gave  $36 
more  than  B,  who  gave  onlyf  as  much  as  A.  How  much 
money  did  each  give,  and  how  much  did  both  give  ? 

CONSIDERATION. 

Compare  A's  gift  with  B's. 

A's  gift  =  B's  gift  +  $36. 

B's  gift  =  f  x  A's  gift. 
If  B  gave  f  as  much  as  A,  the  latter  gave  |. 


i66 


THE  FOURTH  YEAR. 


How  many  parts  did  A  give,  and  how  many  B  ? 
gave  4  and  B  3  parts.     Both  gave  44-3=7  parts. 

How  many  more  parts  did  A  give  than  B  ? 

What  is  one  part  of  the  whole  ?     (-f .) 

How  many  dollars  did  A  give  more  than  B  ? 

Then  $36  =  what  part  of  the  whole  ?     (|.) 

Since  $36  =  -f,  how  much  is  the  whole  or  £? 

(7  x  $36  =  $252.) 

What  part  of  this  did  A  give,  and  what  part  B  ? 

A  and  B  gave  together  $252,  of  which  A  gave 
how  much  did  B  give  ? 

B  gave  $108 ;  how  much  did  A  give? 

A  gave  $144,  B  $36  less. 

That  which  A  gave  more  than  B  was  ^  of  the  whole; 
what  was  the  whole  ? 

FOURTH  STEP. 

Fifths. 

1. 


i  i  i  i 

5  in  i  =  \,  or  \  x  i  =  £ 
Proceed  as  in  the  former  steps. 
COMPARING. 

a.  4-  with  4% 


I 

2 

3 

4| 

5 

6 

7 

8 

9 

ID!  ii 

12 

13 

14 

is'  '16 

I" 

18 

19 

20 

Compare  \  with  £ ,  f  with  J,  f  with  f . 
(Allow  the  pupils  to  illustrate  these  comparisons  on  the 
blackboard.) 


FIXST  HALF  OF   THE   VEAR. 


167 


b.      with     . 


II       ij      13      14      1 


Compare  £  with  f. 


etc. 

c.  What  is  the  common  denominator  of  halves,  thirds, 
fourths,  and  fifths  ? 

(Halves,  thirds,  and  fourths  have  a  common  denomina- 
tor in  12.  If  I  divide  12  by  fifths  or  5  by  twelfths,  I 
get  sixtieths.  So  i  =  f ' 


Since 

-5-  10  X  J 

Since  f  - 


=  f,  6  xJ-r-6  x  £= 


=|  ;   10  x 


3  =  3  x  i. 
5  =  5x1. 

As  i  =  £  of  5,  3  is  f  x  5  less  than  5. 

As  i  =  i  of  3,  5  is  4  greater  than  3. 

The  5  has  5  of  the  same  kind  of  parts  of  which  the  3 
has  3 ;  therefore  5  =  ^x3  and  3  =  1x5. 

The  relation  occurs  with  6  and  10  (2  x  3  and  2  x  5),  9 
and  15,  12  and  20,  etc. 


1 68  THE  FOURTH  YEAR. 

In  the  same  manner  find  the  relation  of  4  and  5,  and 
their  multiples. 

i  dozen  =  £  of  15,  15  =  £  of  i  dozen. 
e.  Two  numbers,  of  which  one  is6J,  have  a  sum  of  i8f. 
What  is  the  other  number  ? 

18  —  6  =  12,  £  —  £  =  }£  —  ^  =  -A-.      r8£  __ 
Or:    i8f  —  6  =  12*,  12*  —  £  = 


/.  How  many  times  must  I  take  3f  to  get  18  ? 

(3|  =  y,  18  =  V ;  Y  ^  ¥  =  18  in  90  =  5.) 

3.  How  do  I  get  £  of  a  cwt.  ? 

(£  of  i  cwt.  taken  4  times.) 

Express  the  difference  between  \  cwt.  and  $•  cwt.  in 
pounds,  i  and  J.  \  and  J.  £  and  £.  £  and  J,  etc. 

How  many  dimes  must  I  add  to  \  of  a  dollar  to  get  £  of 
a  dollar  ? 

If  £  Ib.  costs  i  dollar,  how  much  will  \  lb.  cost  ? 

(Since  \  lb.  costs  i  dollar,  i  lb.  will  cost  3  x  i  =  f  dol- 
lar. £  lb.  will  cost  then  \  x  f  dollar  =  ^  dollar. 

If  a  man  applies  f  of  his  income  for  his  support,  \  of 
the  remainder  for  pleasure,  and  has  $48  left,  how  much  is 
his  income? 

(As  he  spends  f  for  his  support,  there  remains  \.  He 
spends  J  of  the  remainder  [i  x  -J-]  -^  for  pleasure.  Thus 
he  spends  f  +  TV  =  TIT  +  iV  =  &  and  retains  T4F.  Then 
T4F  =  $48,  A  =  $12  and  if  =  $180.) 

How  much  money  does  his  support  cost  him?  His 
pleasure  how  much?  What  relation  do  these  amounts 
bear  to  each  other  ?  What  relation  between  the  money 
for  pleasure  and  that  spared  ?  What  part  of  the  whole 
is  the  $48  ? 

A  person  has  an  income  of  $180.  He  spends  f  for  his 
support  and  -fa  for  pleasure.  How  much  does  he  save  ? 

A  man  has  an  income  of  $180,  f  of  which  is  necessary 
for  his  support,  and  $48  is  saved.  How  much  and  what 
part  does  he  spend  for  pleasure  ? 


FIRST  HALF  OF   THE   YEAR.  169 

FIFTH  STEP. 
Sixths. 

1. 
i  i 


^^J^^^^ 

T     ~T      T" 

As  in  preceding  steps : 

i  x  |  =  |,  2  x  J  =  i  3  x  J  =  fc  etc. 

i  -r-  f  =  6,'  i  -f-  f*=  3!  1+1  =  2,  etc! 

a.  As  6  parts  =2x3  or  3x2  parts,  sixths  can  be  ex- 
pressed in  thirds  and  halves.     As  8  parts  =  2  x  4  or  4  x 
2  parts,  eighths  can  be  expressed  in  halves  and  fourths. 
In  the    same  way  twelfths  can  be  reduced  to  halves, 
thirds,  fourths,  and  sixths. 

Why  cannot  £  be  expressed  in  halves  or  thirds?  (Be- 
cause £  is  not  contained  in  £  without  remainder.) 

b.  Compare  \  and  J. 

(Pupil  will  now  be  able  to  do  this  without  help.) 
Compare  \  with  J  ? 

(Pupil  resolves  both  into  24ths,  and  soon  sees  that  I2ths 
are  still  more  simple. 

c.  What  is  the  common  denominator  of  halves,  thirds, 
fourths,  and  sixths  ? 

What  is  the  difference  between  f  and  |  ?    f  and  £  ? 

What  relation  exists  between  5  and  6? 

(5=1  x  6,6  =  f  x  5.) 

Show  that  |  of  5  and  £  of  6  hold  the  same  relation  as  5 
and  6  do.  Also  that  2  times  5  and  2  times  6  hold  the 
same  relation  that  5  and  6  do. 

1  dime    =  ^  of  a  dollar. 

2  dimes  =  ^  =  ^  of  a  dollar. 

3  dimes  =  -f^  of  a  dollar. 

4  dimes  =  T%  =  f  of  a  dollar. 

etc. 


17°  THE  FOURTH  YEAR. 

1  oz.  =  T\  of  a  pound. 

2  oz.  =  T\  =  £  of  a  pound. 

3  oz.  =  T\  of  a  pound. 

4  oz.  =  r\  =  J  of  a  pound. 

etc. 

In  the  same  manner  make  practical  applications  of  frac- 
tions in  connection  with  compound  numbers  and  in  daily 
life.  These  five  steps  will  be  sufficient  to  illustrate  the 
method.  The  teacher  will  easily  lead  the  pupils  step  by 
step,  until  they  are  able  to  do  the  work  alone.  Many 
examples  must  be  given  in  order  that  the  pupils  become 
thorough  and  efficient.  This  work  will  employ  the  first 
half  of  the  year. 


SECOND  HALF  OF  THE  YEAR. 

THE  FOUR  FUNDAMENTAL  RULES  IN 
FRACTIONS. 

The  pupils  have  been  thus  far  taught  to  consider  the 
fraction  from  all  sides.  It  remains  now  to  take  each 
operation  by  itself,  and  bring  the  knowledge  of  it  to  com- 
pletion. Let  it  not  be  forgotten  that  accuracy  and  rapid- 
ity must  be  secured  before  the  pupil  can  be  said  to  have 
reached  complete  mastery  of  any  step. 

CLASSIFICATION. 

1.  Nature  and  manner  of  treating  the  fraction  in  gene- 
ral (explanation  of  the  parts,  kinds,  etc.,  amplifying,  com- 
mon denominator,  etc.) 

2.  Resolution  (resolving).  1 


3.  Reduction. 

4.  Addition. 

5.  Subtraction. 

6.  Multiplication, 

7.  Division, 


In    abstract    and    concrete 
numbers  pral  and  written. 


SECOND  HALF  OF   THE   YEAR.  IJl 

There  will  be  no  difficulty  in  connection  with  these 
steps,  if  the  preceding  have  been  carefully  taught.  Grube 
calls  especial  attention  to  the  following  points : 

I.   The  Unity. 

The  pupil  has  already  learned  that  when  i  is  divided 
into  4  equal  parts,  one  part  =  \. 

Three  fourths  are  3  of  the  4  parts,  into  which  i  has 
been  divided. 

The  unity  may  be  of  any  size  or  number  which  is 
divided  into  equal  parts,  as :  a  yard,  a  cwt.,  a  ton,  etc. 

Draw  J  of  a  yard,  f  of  a  yard. 


In  order  to  get  b  what  must  I  first  have  ?    (a.) 
What  is  a  ?     (The  whole  or  unity.) 
What  quantity  in  the  figure  constitutes  the  unity  ? 
(The  yard.) 

If  we  consider  a,  i  cwt.,  as  unity,  how  much  does  b 
equal  ?    c  ?    (£  cwt.,  f  cwt.) 

How  many  pounds  would  b  equal  ?    c  ? 

If  b  =  25  Ibs.,  how  much  is  the  unity  or  whole  ? 

If  b  =  27!  Ibs.,  how  much  is  the  unity  ? 

27^  Ibs. 


The  number  with  which  I  indicate  the  unity  is  i. 

What  do  we  call  the  numbers  which  indicate  parts  of 
the  unity? 

What  is  i?    f?    i?     (A  fraction.) 

What  then  is  a  fraction  ?  (A  fraction  is  one  or  more 
of  the  equal  parts  of  a  unity.) 

How  many  parts  have  we  in  £  ?     (We  have  i  part.) 

In  f  ?     (We  have  3  parts.) 


I72  THE  FOURTH  YEAR. 

Into  what  is  the  unity  divided  when  we  get  \  ?  (Into  3 
parts  or  thirds.) 

When  we  get  |  ?     (Into  6  parts  or  sixths.) 

What  do  we  call  the  6  or  the  number  below  the  line  ? 

(The  denominator.) 

What  do  we  call  the  5  or  the  number  above  the  line  ? 

(The  numerator.) 

The  teacher  must  give  many  more  questions  similar  to 
the  above,  until  the  pupils  understand  thoroughly  the 
meaning  of  the  terms  fraction,  numerator,  and  denomi- 
nator. 

How  does  the  fraction  £  compare  with  unity  ? 

(It  is  less  than  unity.) 

The  fraction  £?    (It  is  less  than  unity.) 

A  fraction  which  is  less  than  unity,  we  call  a  proper 
fraction. 

How  does  a  proper  fraction  compare  with  unity  ?  (It 
is  less.) 

(Many  more  similar  questions.) 

What  fraction  equals  the  unit?  (That  which  takes  all 
the  parts  into  which  the  unit  is  divided.) 

Express  the  unity  in  thirds,  fourths,  tenths,  thou- 
sandths. 

(The  transition  from  proper  to  improper  fractions  is 
easy.) 

2.  Expansion  and  Reduction. 

If  you  multiply  f  by  3,  what  do  you  get  ? 
Illustrate  this. 


SECOND  HALF  OF   THE   YEAR.  173 

Change  J  into  whole  numbers. 


I 1* 

Which  term  of  the  fraction  have  I  multiplied,  and  which 
remains  unchanged  in  expanding  f  ? 

(Numerator  has  been  multiplied,  denominator  un- 
changed.) 

If  I  multiply  the  numerator  by  3,  5, 10,  etc.,  what  of  the 
value  of  the  fraction  ? 

(3,  5,  10,  etc.,  times  greater.) 

Leaving  the  numerator  unchanged  and  multiply  the 
denominator  by  3,  what  fraction  do  we  get  ?  (T3^.) 

How  does  f  compare  with  T%?     (It  is  3  times  as  large.) 

If  I  multiply  the  denominator  4  by  3,  it  is  the  same  as 
to  take  what  part  of  the  fraction  ? 

Illustrate  this. 


If  f  be  divided  by  3  we  get  T%  =  i. 

What  must  be  done  with  -£%  or  J  to  get  again  the  first 
value  or  f  ? 

What  effect  upon  the  value  of  a  fraction  if  both  numer- 
ator and  denominator  are  multiplied  by  the  same  number? 

Continue  work  of  this  kind  until  the  pupils  are  familiar 
with  all  the  changes,  such  as  multiplying  or  dividing  th 
numerator  or  denominator,  or  both,  etc.  Illustrate  each 
step  in  the  manner  already  indicated.  This  will  include 
reducing  to  lower  terms,  to  higher  terms,  to  fractions 
having  a  given  denominator,  improper  fractions  to  mixed 
numbers,  mixed  numbers  to  improper  fractions,  etc. 


174  THE  FOURTH  YEAR. 

3.  Common  Denominator. 

It  is  necessary  in  adding  fractions  to  find  a  common 
denominator,  that  is,  a  general  denominator  into  which 
the  other  denominators  will  go  without  a  remainder.  It 
must  also  the  smallest  number  into  which  all  the  de- 
nominators will  go  without  a  remainder.  This  can  be 
found  by  using  the  prime  factors  of  the  numbers  whose 
common  denominator  we  seek.  Take  J  and  \. 

4  =  2x2 
6x3x2 

Both  the  4  and  6  have  the  common  factor  2,  and  this 
is  taken  but  once.  We  then  have  3x2x2  =  12.  Since 
the  factors  of  6  (3  x  2),  and  those  of  4  (2  x  2)  are  found 
in  the  factors  of  12  (3  x  2  x  2),  12  will  contain  6  and  4, 
and  is  their  common  denominator — it  is  also  the  common 
denominator  of  J  =  T3^-,  and  \  =  T%. 

Take  fc,  A,  *. 

15  =  5  x  3 

18  =  2  x  3  x  3 

9  =  3><3 

Since  9  is  contained  in  18,  it  is  also  contained  in  a  mul- 
tiple of  1 8.  Therefore  the  common  denominator  of  15 
and  1 8  will  also  contain  9;  so  we  will  consider  15  and  18. 

15  and  18  have  the  common  factor  3.  In  the  factors  5  x 
3x3  x  2  we  find  all  the  factors  of  15  and  18;  therefore 
their  product,  or  90,  is  a  common  dividend  for  both  and 
also  for  9. 

Give  other  examples. 

Add  the  following  fractions : 

M  +  «  +  A  +  T!*  +  *. 

Write  the  denominators  in  a  column,  placing  at  the  left 
a  small  prime  factor  which  will  be  contained  in  two  or 
more  of  them : 

f  36  =  12x3 

!   is-  5x3 
3\    n  =  n 

105  =  35x3 
I     3=    1x3 


SECOND  HALF  OF  THE   YEAR. 


175 


But  there  is  another  prime  factor  (5)  which  is  contained 
in  more  than  one  of  the  numbers  : 

136= 
15=    3 
ii  =  u 
105  =  35  = 
3  =    i  >< 

We  have  then  remaining  the  factors  12,  3,  n,  7,  5. 
Or  it  may  be  expressed  in  a  shorter  way  :* 


5 


36,  15,  11,  105,  3 


12,    5,  n,    35,  i 


12,    i,  11, 


4.  Number  Relations  in  the  Fractional  Form. 

.     What  relations  exists  between  the  numbers  5  and  9  ? 

If  I  wish  to  measure  two  numbers,  I  must  measure 
them  with  one  another  by  the  same  measure.  But  5  and 
9  have  no  common  measure  besides  the  i.  As  5  =  5  x  i, 
i  must  equal  i  x  5  with  reference  to  the  5,  and  \  x  9  with 
reference  to  the  9.  Therefore  9  in  relation  to  the  5  is 
nine  times  \  of  5,  and  5  is  five  times  \  of  9. 


5  and  9. 


It  follows : 


5  =  5 

i=t 


9  =  9  x  i 
i=|  x  9 


5  =  |  x  9  and 


*  To  assist  in  factoring,  notice  the  following  facts : 

1.  All  numbers  which  end  in  o  are  exactly  divisible  by  2  and  5. 

2.  All  numbers  which  end  in  o  or  5  are  exactly  divisible  by  5. 

3.  All  numbers  which  end  in  2,  4,  6  or  8  are  exactly  divisible  by  2. 

4.  All  numbers  are  exactly  divisible  by  3,  the  sum  of  whose  digits  is 
divisible  by  3.    For  example,  4365,  the  sum  of  the  digits  is  18;  since  18  is 
divisible  by  3,  4365  is  also. 

5.  All  numbers  are  exactly  divisible  by  9,  the  sum  of  whose  digits  is 
divisible  by  9. 

6.  All  numbers  are  divisible  by  6  which  are  divisible  by  2  and  3. 

7.  All  numbers  are  divisible  by  8,  the  sum  of  whose  last  three  figures  is 
divisible  by  8. 


176  THE  FOURTH  YEAR. 

3  and  4. 

3  =  1  X  4 

4  =  |X3. 

In  comparing  3  and  9  with  i  as  the  measure,  I  have 
3  =  |X9,  9  =  1x3.     With  3  as  the  measure,  I  have 
3  =  £  x  9 
9  =  3x3. 

APPLICATION. 

If  5  yards  of  cloth  cost  $4,  what  will  9  yards  cost  ? 

(As  9  yards  =  £  x  5  yards,  they  will  cost  -§-  x  $4  =  &£•  = 
*7iO 

As  every  fraction  is  a  division,  and  every  example  in 
proportion  goes  out  from  a  divisor  and  dividend,  it  is  ad- 
visable to  make  use  (according  to  Grube's  idea)  of  the 
fractional  form  of  expression  in  proportion.  It  has  the 
advantage  that  it  shows  objectively  the  solution  of  the 
example. 

If  5  yards  cost  $4,  what  will  9  yards  cost  ? 

(Suppose  that  i  yard  costs  $4,  9  yards  will  be  worth  9  x 
$4.  It  is  not  i  yard,  but  5  yards  that  cost  $4  ;  therefore 
the  price  will  be  5  times  less. 

9|i  dollars  =  |  =  $7i) 

A  quantity  of  hay  will  last  5  horses  4  days  ;  how  long 
will  it  last  9  horses  ? 

(Suppose  the  hay  was  sufficient  for  i  horse  for  4  days, 
for  9  horses  it  would  last  f  days.  But  as  it  is  sufficient 
for  5  horses  instead  of  i,  it  will  last  5  times  f  days  = 


9  9 

3f  bushels  of  rye  cost  $5  ;  what  will  4  bushels  cost  at  the 
same  rate  ? 

4x5.  20      4  x  20      80 

?-^>-  dollars  =  —  =  i_  =  —  =  5*  dollars. 

4  bushels  cost  $5,  what  will  3f  bushels  cost  ? 

5x3!      5  x  -1/      5  x  15      75 

2.  -  &  =  .2  -  ±  =  £  -  ?  =  LP  —  411  dollars.) 
4  4  4  x  4        16 


SECOND  HALF  OF  THE  YEAR.  1/7 

This  method  is  more  elementary  than  the  usual  method 
of  stating  proportion,  and  it  is  none  the  less  a  practice  in 
thinking. 

The  elementary  school  has  accomplished  enough  in  the 
first  four  years,  if  it  has  brought  the  pupil  to  be  able  to 
solve  simple  practical  examples  rapidly  and  accurately 
either  by  analyzing  back  to  unity,  or  by  means  of  the 
fractional  method,  or  by  comparing  the  relations.  He 
will  also  be  able  to  decide  by  his  own  observation  which 
method  of  solution  is  the  best  for  any  given  example.  A 
pupil,  who  knows  95  to  equal  5  x  19  will  easily  solve 
such  an  example  as  the  following  :  If  6£  Ibs.  of  flour  cost 
20  cents,  what  will  95  Ibs.  cost  ? 

(6i  =  ^.  As  95  =  5  x  19,  95  Ibs.  =  15  x  ^  pounds. 
^  Ibs.  cost  20  c.,  and  15  x  20  c.  =  300  c. 

Analyzing  back  to  unity  : 

6i  Ibs  ...........  20  c. 


19 


95  "    ..........  285  +  15  =  300  c.) 

If  the  teacher  has  faithfully  followed  this  course,  the 
pupils  are  prepared  for  a  practical  arithmetic,  and  they 
should  be  given  a  book  having  a  great  many  examples 
methodically  arranged.  With  such  a  book,  they  will 
be  able  to  solve  and  explain  the  examples  from  be- 
ginning to  end.  The  teacher  must  give  a  great  many 
original  examples.  On  the  other  hand,  he  must  not  feel 
bound  to  use  all  of  the  examples  in  this  book  with  which 
to  drill  his  pupils.  He  must  use  judgment  and  common 
sense  in  the  application  of  the  principles  and  methods 
herein  given. 

Not  many  kinds,  but  much,  is  the  motto  with  which 
Grube  closes  his  work  on  Number. 


•TjmBRSITYI 


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25 


'Payne  s  Lectures  on   the  Science   and 

ART  OF  EDUCATION.  Reading  Circle  Edition.  By  JOSEPH 
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Philadelphia  Educational  News.—"  Ought  to  be  in  library  of  every 
progressive  teacher." 

Educational  Courant.— "  To  know  how  to  teach,  more  i?  needed  than 
a  knowledge  of  the  braiicnes  taught.  This  is  especially  valuable." 

Pennsylvania  Journal  of  Education.—"  Will  be  of  practical  value  tc 
Normal  Schools  and  Institute 


SEND  ALL  ORDERS  TO 

E.  L.  KELLOGG  &  CO.,  NEW  YORK  &  CHICAGO.  47 


Welcb's  Teachers  Psychology. 

A  Treatise  on  the  Intellectual  Faculties,  the  Order  of  the 
Growth,  and  the  Corresponding  Series  of  Studies  by  which 
they  are  Educated,  By  the  late  A.  S.  Welch,  Professor  of 
Psychology,  Iowa  Agricultural  College,  formerly  Pres.  of 
the  Mich.  Normal  School.  Cloth,  12mo,  300  pp.,  $1.25;  to 
teachers,  $1;  by  mail,  12  cents  extra.  Special  terms  to 
Kormal  Schools  and  Reading  Circles. 

A  mastery  of  the  branches  to  be  taught  was  once  thought  to  be 
an  all-sufficient  preparation  for  teaching.  But  it  is  now  seen  that 
there  must  be  a  knowledge  of  the  mind  that  is  to  be  trained. 
Psychology  is  the  foundation  of  intelligent  pedagogy.  Prof. 
Welch  undertook  to  write  a  book  that  should  deal  with  mind- 
unfolding,  as  exhibited  in  the 
school-room.  He  shows  what  is 
meant  by  attending,  memorizing, 
judging,  abstracting,  imagining, 
classifying,  etc.,  as  it  is  done  by 
the  pupil  over  his  text-books.  First, 
there  is  the  concept;  then  there  is 
(1)  gathering  concepts,  (2)  storing 
concepts,  (3)  dividing  concepts, 
(4)  abstracting  concepts,  (5)  build- 
ing concepts,  (6)  grouping  con- 
cepts, (7)  connecting  concepts, 
(8)  deriving  concepts.  Each  of 
these  is  clearly  explained  and  il- 
lustrated ;  the  reader  instead  of 
being  bewildered  over  strange 
terms  comprehends  that  imagina- 
tion means  a  building  up  of  con- 
cepts, and  so  of  the  other  terms. 
A  most  valuable  part  of  the  book 
is  its  application  to  practical  education.  How  to  train  these 
powers  that  deal  with  the  concept — that  is  the  question.  There 
must  be  exercises  to  train  the  mind  to  gather,  store,  divide,  abstract, 
build,  group,  connect,  and  derive  concepts.  The  author  shows 
what  studies  do  this  appropriately,  and  where  there  are  mistakes 
made  in  the  selection  of  studies.  The  book  will  prove  a  valuable 
one  to  the  teacher  who  wishes  to  know  the  structure  of  the  mind 
and  the  way  to  minister  to  its  growth.  It  would  seem  that^at 
last  a  psychology  had  been  written  that  would  be  a  real  aid,  in- 
stead of  a  hindrance,  to  clear  knowledge. 


DR.  A.  S.  WELCH. 


SEND  ALL  ORDERS  TO 

E.  L.  KELLOGG  &  CO.,  NEW  YORK  &  CHICAGO. 


Aliens  Mind  Studies  for  Young  Teacb- 

BBS.  By  JEROME  ALLEN,  Ph.D.,  Associate  Editor  of  the 
SCHOOL  JOURNAL,  Prof,  of  Pedagogy,  Univ.  of  City  of 
N.  Y.  16mo,  large,  clear  type,  128  pp.  Cloth,  50  cents ;  to 
teachers,  40  cents ;  by  mail,  5  cents  extra. 

There  are  many  teachers  who 
know  little  about  psychology, 
and  who  desire  to  be  better  in- 
formed concerning  its  princi- 
ples, especially  its  relation  to  the 
work  of  teaching.  For  the  aid 
of  such,  this  book  has  been  pre- 
pared. But  it  is  not  a  psychol- 
ogy— only  an  introduction  to  it, 
aiming  to  give  some  funda- 
mental principles,  together  with 
something  concerning  the  phi- 
losophy of  education.  Its  meth- 
od is  subjective  rather  than  ob- 
jective, leading  the  student  to 
watch  mental  processes,  and 
draw  his  own  conclusions.  It 
is  written  in  language  easy  to 
be  comprehended,  and  has  many 
JEROME  ALLEN,  Ph.D., Associate  Editor  Practical  illustrations.  It  will 
of  the  Journal  and  institute.  aid  the  teacher  in  his  daily  work 
in  dealing  with  mental  facts  and  states. 

To  most  teachers  psychology  seems  to  be  dry.  This  book  shows 
how  it  may  become  the  most  interesting  of  all  studies.  It  also 
shows  how  to  begin  the  knowledge  of  self.  "  We  cannot  know 
in  others  what  we  do  not  first  know  in  ourselves."  This  is  the 
key-note  of  this  book.  Students  of  elementary  psychology  will 
appreciate  this  feature  of  "Mind  Studies." 
ITS  CONTENTS. 

CHAP. 

I.  How  to  Study  Mind. 
II.  Some  Facts  in  Mind  Growth. 

III.  Development. 

IV.  Mind  Incentives. 

V.  A  few  Fundamental  Principles 

Settled. 

VI.  Temperaments. 
VII.  Training  of  the  Senses. 
VIII.  Attention. 
IX.  Perception. 
X.  Abstraction. 

XI.  Faculties     used     in    Abstract 
Thinking. 


CHAP. 

XII.  From  the  Subjective  to  the 
Conceptive. 

XIII.  The  Will. 

XIV.  Diseases  of  the  Will. 
XV.  Kinds  of  Memory. 

XVI.  The  Sensibilities. 
XVII.  Relation  of  the  Sensibilities 

to  the  Will. 

XVIII.  Training  of  the  Sensibilities. 
XIX.  Relation  of  the  Sensibilities! 

to  Morality. 
XX.  The  Imagination. 
XX£  Imagination  in  its  Maturity. 
XXII.  Education  of  the  Moral  Sence. 


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32    E.  L.  KELLOGG  <6  GO.,  NEW  YORK  &  CHICAGO. 

First  Three  Years  of  Childhood. 

AN  EXHAUSTIVE  STUDY  OP  THE  PSYCHOLOGY  OF  CHILDREN.  By 
BERNARD  PEREZ.  Edited  and  translated  by  ALICE  M.  CHRISTIE, 
translator  of  "  Child  and  Child  Nature,"  with  an  introduction  by 
JAMES  SULLY,  M.A.,  author  of  "  Outlines  of  Psychology,"  etc. 
12mo,  cloth,  324  pp.  Price,  $1.50 ;  to  teachers,  $1.20 ;  by  mail,  10 
cents  extra. 

This  is  a  comprehensive  treatise  on  the  psychology  of  childhood,  and 
is  a  practical  study  of  the  human  mind,  not  full  formed  and  equipped 
with  knowledge,  but  as  nearly  as  possible,  ab  origine — before  habit, 
environment,  and  education  have  asserted  their  sway  and  made  their 
permanent  modifications.  The  writer  looks  into  all  the  phases  of  child 
activity.  He  treats  exhaustively,  and  in  bright  Gallic  style,  of  sensa- 
tions, instincts,  sentiments,  intellectual  tendencies,  the  will,  the  facul- 
ties of  aesthetic  and  moral  senses  of  young  children.  He  shows  how 
ideas  of  truth  and  falsehood  arise  in  little  minds,  how  natural  is  imita- 
tion and  how  deep  is  credulity.  He  illustrates  the  development  of  im- 
agination and  the  elaboration  of  new  concepts  through  judgment, 
abstraction,  reasoning,  and  other  mental  methods.  It  is  a  book  that 
has  been  long  wanted  by  all  who  are  engaged  in  teaching,  and  especially 
by  all  who  have  to  do  with  the  education  and  training  of  children. 

This  edition  has  a  new  index  of  special  value,  and  the  book  is  care- 
fully printed  and  elegantly  and  durably  bound.  Be  sure  to  get  our 
standard  edition. 

OUTLINE  OF  CONTENTS. 

CHAP. 

IX.  Association  of  Psychical  States 

— Association — Imagination. 
X.  Elaboration  of  Ideas— Judg- 
ment —  Abstraction  —  Com- 
parison —  Generalization  — 
Reasoning — Errors  and  Allu- 
sions—Errors and  Allusions 
Owing  to  Moral  Causes. 

XI.  Expression  and  Language. 
XII.  Esthetic        Senses  —  Musical 
Sense  —  Sense    of     Material 
Beauty  —  Constructive      In- 
stinct— Dramatic  Instinct. 
Xni.  Personalty  — Reflection— Moral 
Sense. 


CHAP. 

I.  Faculties  of  Infant  before  Birth 
—First  Impression  of  New- 
born Child. 

II.  Motor  Activity  at  the   Begin- 
ning of  Life— at  Six  Months— 
—at  Fifteen  Months. 
HI.  Instinctive  and  Emotional  Sen- 
sations— First  Perceptions. 
IV.  General  and  Special  Instincts. 
V.  The  Sentiments. 
VI.  Intellectual    Tendencies— Ver- 
acity—Imitation— Credulity. 
VII.  The  Will. 

VIII.  Faculties  of  Intellectual  Acqui- 
sition and  Retention— Atten- 


tion— Memory. 

Col.  Francis  W.  Parker.  Principal  Cook  County  Normal  and  Training 
School,  Chicago,  says:— "I  am  glad  to  see  that  you  have  published  Perez's 
wonderful  work  upon  childhood.  I  shall  do  all  I  can  to  get  everybody  to  read 
it.  It  is  a  grand  work." 

John  Bascom,  Pres.  Univ.  of  Wisconsin,  says:—"  A  work  of  marked 
interest." 

CK  Stanley  Hall,  Professor  of  Psychology  and  Pedagogy,  Johns  Hopkins 
Jniv.,  says: — "  I  esteem  the  work  a  very  valuable  one  f 


Univ.,  says:— "I  esteem  the  work  a  very  valuable  one  for  primary  and  kin- 
dergarten teachers,  and  for  all  interested  in  the  psychology  of  childhood." 
And  many  other  strong  commendations. 


BEND  AM,  ORDERS  TO 

B.  L.  KELLOGG  &  CO.,  NEW  YORK  &  CHICAGO.     27 

Parkers  Talks  on  Teaching. 

Notes  of  "Talks  on  Teaching"  given  by  COL.  FRANCIS  W. 
PARKER  (formerly  Superintendent  of  schools  of  Quincy, 
Mass.),  before  the  Martha's  Vineyard  Institute,  Summer 
of  1882.    Reported  by  LEIJA  E.  PATRIDGE.    Square  16mo, 
5x6  1-2  inches,  192  pp. ,  laid  paper,  English  cloth.    Price, 
$1.25  ;  to  teachers,  $1.00  ;  by  mail,  9  cents  extra. 
The  methods  of  teaching  employed  in  the  schools  of  Quincy, 
Mass.,  were  seen  to  be  the  methods  of  nature.    As  they  were 
copied  and  explained,  they  awoke  a  great  desire  on  the  part 
of  those  who  could  not  visit  the  schools  to  know  the  underly- 
ing principles.    In  other  words,  Colonel  Parker  was  asked  to 
explain  why  he  had  his  teachers  teach  thus.    In  the  summer 
of  1882,  in  response  to  requests,  Colonel  Parker  gave  a  course 
of  lectures  before  the  Martha's  Vineyard  Institute,  and  these 
were  reported  by  Miss  Patridge,  and  published  in  this  book. 

The  book  became  famous  ; 
more  copies  were  sold  of  it  in 
the  same  time  than  of  any 
other  educational  book  what- 
ever.  The  daily  papers,  which 
usually  pass  by  such  books 
with  a  mere  mention,  devoted 
columns  to  reviews  of  it. 
The  following  points  will 


show   why  the   teacher  will 
want  this  book. 

1.  It  explains   the   "  New 
Methods."     There  is  a  wide 
gulf  between  the  new  and  the 
old  education.     Even   school 
boards  understand  this. 

2.  It  gives  the  underlying 
principles  of  education.  For  it 

must  be  remembered  that  Col.  Parker  is  not  expounding  his 
methods,  but  the  methods  of  nature. 

3.  It  gives  the  ideas  of  a  man  who  is  evidently  an  "  educa- 
tional genius,"  a  man  born  to  understand  and  expound  educa- 
tion.   We  have  few  such  ;  they  are  worth  everything  to  the 
human  race. 

4.  It  gives  a  biography  of  Col.  Parker.    This  will  help  the 
teacher  of  education  to  comprehend  the  man  and  his  motives. 

0.  It  has  been  adopted  by  nearly  every  State  Reading  Circle. 


SEND  ALL  ORDERS  TO 

18  E.  L.  KELLOGG  &  CO.,  NEW  YOBK  &  CHICAGO. 


Hughes  ^Mistakes  in  Teaching. 

BY  JAMES  J.  HUGHES,  Inspector  of  Schools,  Toronto,  Canada. 
Cloth,  16mo,  115  pp.  Price,  50  cents;  to  teachers,  40  cents; 
by  mail,  5  cents  extra. 

Thousands  of  copies  of  the  old 
edition  have  been  sold.     The  new 
edition  is  worth  double  the  old; 
the  material  has  been  increased, 
restated,    and    greatly    improved. 
Two  new  and  important  Chapters 
have  been  added  on  "Mistakes  in 
Aims,"  and   "Mistakes  in  Moral 
Training."    Mr.  Hughes  says  in  his 
preface:  "In  issuing  a  revised  edi- 
tion of  this  book,  it  seems  fitting  to 
acknowledge  gratefully  the  hearty 
appreciation  that  has  been  accorded 
it  by  American  teachers.     Realiz- 
ing as  I  do  that  its  very  large  sale 
.  indicates  that  it  has  been  of  service 
;  to  many  of  my  fellow- teachers,  I 
\  have  recognized  the  duty  of  enlarg- 
ing and  revising  it  so  as  to  make  it 
still  more  helpful    in    preventing 
JAMES  L.  HUGHES,  Inspector  of  the  common  mistakes  in  teaching 
Schools,  Toronto,  Canada.         an(j  training. " 

This  is  one  of  the  six  books  recommended  by  the  N.  Y.  State 
Department  to  teachers  preparing  for  examination  for  State  cer* 
tificates. 

CAUTION. 

Our  new  AUTHORIZED  COPYRIGHT  EDITION,  entirely  rewritten  by 
tJie  author,  is  the  only  one  to  buy.  It  is  beautifully  printed  and 
handsomely  bound.  Get  no  other. 

CONTENTS  OF  OUR  NEW  EDITION. 

CHAP.     I.    7  Mistakes  in  Aim. 
CHAP.    II.  21  Mistakes  in  School  Management. 
CHAP.  III.  24  Mistakes  in  Discipline. 
CHAP.  IV.  27  Mistakes  in  Method. 
CHAP.    V.  13  Mistakes  in  Moral  Training. 
'  Chaps.  I.  and  V.  are  entirely  new. 


SEND  ALL  ORDERS  TO 

£0    K  L.  KELLOGG  &  GO.,  NEW  YORK  &  CHICAGO. 

Hughes'  Securing  and  Retaining  Atten* 

TION.  By  JAMES  L.  HUGHES,  Inspector  Schools,  Toronto, 
Canada,  author  of  "  Mistakes  in  Teaching."  Cloth,  116pp. 
Price,  50  cents;  to  teachers,  40  cents;  by  mail,  5  cents  extra. 

This  valuable  little  book  has  already  become  widely  known  to 
American  teachers.  Our  new  edition  has  been  almost  entirely 
re-written,  and  several  new  important  chapters  added.  It  is  the 
only  AUTHORIZED  COPYRIGHT  EDITION.  Caution. — Buy  no  other. 

WHAT    IT   CONTAINS. 

I.  General  Principles;  II.  Kind*  of  Attention;  III.  Characteristics  of  Good 
Attention;  IV.  Conditions  of  Attention;  V.  Essential  Characteristics  of  the 
Teacher  in  Securing  and  Retaining  Attention ;  VI.  How  to  Control  a  Class; 
VII.  Methods  of  Stimulating  and  Controlling  a  Desire  for  Knowledge;  VIII. 
How  to  Gratify  and  Develop  the  Desire  for  Mental  Activity;  IX.  Distracting 
Attention;  X.  Training  the  Power  of  Attention;  XI.  General  Suggestions 
regarding  Attention. 

TESTIMONIALS. 

S.  P.  Bobbins,  Pres.  McGill  Normal  School,  Montreal,  Can.,  writes  to  Mr. 
Hughes:— "It  is  quite  superfluous  for  me  to  say  that  your  little  books  are 
admirable.  I  was  yesterday  authorized  to  put  the  *  Attention '  on  the  list 
of  books  to  be  used  in  the  Normal  School  next  year.  Crisp  and  attractive 
in  style,  and  mighty  by  reason  of  its  good,  sound  cominon-senso,  it  is  a 
book  that  every  teacher  should  know." 

Popular  Educator  (Boston):—"  Mr.  Hughes  has  embodied  the  best  think- 
ing of  his  life  in  these  pages." 

Central  School  Journal  (la.).— "  Though  published  four  or  five  yeara 
since,  this  book  has  steadily  advanced  in  popularity." 

Educational  Courant  (Ky.).— "It  is  intensely  practical.  There  isn't  a 
mystical,  muddy  expression  in  the  book." 

Educational  Times  (England).—"  On  an  important  subject,  and  admir* 

ably  executed." 

School  Guardian  (England).—"  We  unhesitatingly  recommend  it." 
New  England  Journal  of  Education.—"  The  book  is  a  guide  and  a 

manual  of  special  value." 

New  York  School  Journal.— "Every  teacher  would  derive  benefit  from 
reading  this  volume." 

Chicago  Educational  Weekly.—"  The  teacher  who  aims  at  best  suc^ 
cess  should  study  it." 

Phil.  Teacher.—"  Many  who  have  spent  months  in  the  school-room  would 
be  benefited  by  it." 

Maryland  School  Journal.—"  Always  clear,  never  tedious." 

Va.  Ed.  Journal. — "  Excellent  hints  as  to  securing  attention." 

Ohio  Educational  Monthly.—"  We  advise  readers  to  send  for  a  copy." 

Pacific  Home  and  School  Journal.—"  An  excellent  little  manual." 

Prest.  James  H.  Hoose,  State  Normal  School,  Cortland,  N.  Y.,  says:— 

"  The  book  must  prove  of  great  benefit  to  the  profession." 
Supt.  A.  W.  Edson.  Jersey  City,  N.  J.,  says:—"  A  good  treatise  has  long 

been  needed,  and  Mr.  Hughes  has  supplied  the  want." 


SEND  ALL  ORDERS  TO 

10    E.  L.  KELLOGG  &  CO.,  NEW  YORK  &  CHICAGO. 


Calkins    Ear  and  Voice   Training  by 

MEANS  OF  ELEMENTARY  SOUNDS  OF  LANGUAGE.    By  K.  A. 
CALKINS,   Assistant    Superintendent   N.   Y.   City  Schools ; 
author  of  "Primary  Object  Lessons,"  "Manual  of  Object 
Teaching,"  "  Phonic  Charts,"  etc.     Cloth.     16mo,  about  100 
pp.   Price,  50  cents;  to  teachers,  40  cents;  by  mail,  5  cents  extra. 
An  idea  of  the  character  of  this  work  may  be  had  by  the  fol- 
lowing extracts  from  its  Preface  : 

u  The  common  existence  of  abnormal  sense  perception  among  school 
children  is  a  serious  obstacle  in  teaching.  This  condition  is  most 

obvious  in  the  defective  perceptions 
of  sounds  and  forms.  It  may  be 
seen  in  the  faulty  articulations  in 
speaking  and  reading;  in  the  ina- 
bility to  distinguish  musical  sounds 
readily ;  also  in  the  common  mis- 
takes made  in  hearing  what  is 
said.  .  .  . 

"Careful  observation  and  long 
experience  lead  to  the  conclusion 
that  the  most  common  defects  in 
sound  perceptions  exist  because  of 
lack  of  proper  training  in  childhood 
to  develop  this  power  of  the  mind 
into  activity  through  the  sense  of 
hearing.  It  becomes,  therefore,  a 
,  matter  of  great  importance  in  edu- 
cation, that  in  the  training  of  chil- 
dren due  attention  shall  be  given  to 
the  development  of  ready  and  accu- 
rate perceptions  of  sounds. 

"  How  to  give  this  training  so  as 
to  secure  the  desired  results  is  a 
subject  that  deserves  the  careful 
attention  of  parents  and  teachers. 
Much  depends  upon  the  manner  of 
presenting  the  sounds  of  our  language  to  pupils,  whether  or  not  the 
results  shall  be  the  development  in  sound-perceptions  that  will  trctin 
the  ear  and  voice  to  habits  of  distinctness  and  accuracy  in  speaking  and 
reading. 

"  The  methods  of  teaching  given  in  this  book  are  the  results  of  an 
extended  experience  under  such  varied  conditions  as  may  be  found 
with  pupils  representing  all  nationalities,  both  of  native  and  foreign 
born  children.  The  plans  described  will  enable  teachers  to  lead  their 
pupils  to  acquire  ready  and  distinct  perceptions  through  sense  train- 
ing, and  cause  thein  to  know  the  sounds  of  our  language  in  a  manner 
that  will  give  practical  aid  in  learning  both  the  spoken  and  the  written 
language.  The  simplicity  and  usefulness  of  the  lessons  need  only  to  be 
known  to  be  appreciated  and  used*'7 


SUPT.  N.  A.  CALKINS. 


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E.  L.  KELLOGG  &  CO.,  NEW  YORK  &  CHICAGO. 

Dewey's  How  to  Teach  Manners  in  the 

SCHOOL-ROOM.  By  Mrs.  JULIA  M.  DEWEY,  Principal  of  the 
Normal  School  at  Lowell,  Mass.,  formerly  Supt.  of  Schools 
at  Hoosick  Falls,  N.  Y.  Cloth,  16mo,  104  pp.  Price,  50 
cents;  to  teachers,  40  cents;  by  mail,  5  cents  extra. 

Many  teachers  consider  the  manners  of  a  pupil  of  little  impor- 
tance so  long  as  he  is  industrious.  But  the  boys  and  girls  are  to 
be  fathers  and  mothers;  some  of  the  boys  will  stand  in  places  of 
importance  as  professional  men,  and  they  will  carry  the  mark  of 
ill-breeding  all  their  lives.  Manners  can  be  taught  in  the  school- 
room: they  render  the  school- room  more  attractive;  they  banish 
tendencies  to  misbehavior.  In  this  volume  Mrs.  Dewey  has  shown 
how  manners  can  be  taught.  The  method  is  to  present  some  fact 
of  deportment,  and  then  lead  the  children  to  discuss  its  bearings; 
thus  they  learn  why  good  manners  are  to  be  learned  and  practised. 
The  printing  and  binding  are  exceedingly  neat  and  attractive." 


OUTLINE    OF 

Introduction. 

General  Directions. 

Special  Directions  to  Teachers. 

LESSONS  ON  MANNERS  FOR  YOUNGEST 

PUPILS. 
Lessons  on  Manners  —  Second  Two 

Years. 
Manners  in  School— First  Two  Years. 

Second 
Manners  at  Home— First 

Second 

Manners  in  Public— First 
"  "          Second 


CONTENTS. 

Table  Manners— First  Two  Years. 

Second        *' 
LESSONS  ON  MANNERS  FOR  ADVANCED 

PUPILS. 

Manners  in  School. 
Personal  Habits. 
Manners  in  Public. 
Table  Manners. 
Manners  in  Society. 
Miscellaneous  Items. 
Practical  Training  in  Manners. 
Suggestive    Stories,   Fables,   Anec- 
dotes, and  Poems. 
Memory  Gems. 


Central  School  Journal,—1'  It  furnishes  illustrative  lessons." 
Texas  School  Journal.—"  They  (the  pupils)  will  carry  the  mark  of  ill- 
breeding  all  their  lives  (unless  taught  otherwise)." 

Pacific  Ed.  Journal.—"  Principles  are  enforced  by  anecdote  and  conver- 
sation." 

Teacher's  Exponent.— "  We  believe  such  a  book  will  be  very  welcome." 
National  Educator.—  "  Common-sense  suggestions." 
Ohio  Ed.  Monthly.—"  Teachers  would  do  well  to  get  it." 
Nebraska   Teacher,— "  Many  teachers  consider  manners  of  little  im- 
portance, but  some  of  the  boys  will  stand  in  places  of  importance." 
School  Educator.— "The  spirit  of  the  author  is  commendable." 
School  Herald.—"  These  lessons  are  full  of  suggestions." 
Va.  School  Journal.— "Lessons  furnished  in  a  delightful  style." 
Miss.  Teacher. — "  The  best  presentation  we  have  seen." 
Ed.  Courant.— "  It  is  simple,  straightforward,  and  plain." 
Iowa  Normal  Monthly.—"  Practical  and  well-arranged  lessons  on  man- 
ners." 

Progressive  Educator.—"  Will  prove  to  be  most  helpful  to  the  teacher 
who  desires  her  pupils  to  be  well-mannered." 


fciSND  ALL  ORDMIS  $6 

30     E.  L.  KELLOGG  &  CO.,  NEW  YORK  &  CHICAGO. 


'Quincy  ^Methods" 


The  "  Quincy  Methods,"  illustrated  ;  Pen  photographs  from 
the  Quincy  schools.  By  LELIA  E.  PATRIDGE.  Illustrated 
with  a  number  of  engravings,  and  two  colored  plates. 
Blue  cloth,  gilt,  12mo,  686  pp.  Price,  $1.75  ;  to  teachers, 
$1.40  ;  by  mail,  13  cents  extra. 

When  the  schools  of  Quincy,  Mass.,  became  so  famous 
onder  the  superintendence  of  Col.  Francis  W.  Parker,  thou- 
sands of  teachers  visited  them.  Quincy  became  a  sort  of 
"  educational  Mecca,"  to  the  disgust  of  the  routinists,  whose 
schools  were  passed  by.  Those  who  went  to  study  ths 
methods  pursued  there  were  called  on  to  tell  what  they  had 
seen.  Miss  Patridge  was  one  of  those  who  visited  the  schools 
of  Quincy  ;  in  the  Pennsylvania  Institutes  (many  of  which 
she  conducted),  she  found  the  teachers  were  never  tired  of 
being  told  how  things  were  done  in  Quincy.  She  revisited 
the  schools  several  tunes,  and  wrote  down  what  she  saw  ;  then 
the  book  was  made. 

1.  This  book  presents  the  actual  practice  in  the  schools  of 
Quincy.    It  is  composed  of  "  pen  photographs." 

2.  It  gives  abundant  reasons  for  the  great  stir  produced  by 
the  two  words  "  Quincy  Methods."    There  are  reasons  for  the 
discussion  that  has  been  going  on  among  the  teachers  of  late 
years. 

3.  It  gives  an  insight  to  principles  underlying  real  educa- 
tion as  distinguished  from  book  learning. 

4.  It  shows  the  teacher  not  only  what  to  do,  but  gives  the 
way  in  which  to  do  it. 

5.  It  impresses  one  with  the  spirit  of  the  Quincy  schools. 

6.  It  shows  the  teacher  how  to  create  an  atmosphere  of  hap- 
piness, of  busy  work,  and  of  progress. 

7.  It  shows  the  teacher  how  not  to  waste  her  tune  in  worry' 
ing  over  disorder. 

8.  It  tells  how  to  treat  pupils  with  courtesy,  and  get  cour- 
tesy back  again. 

9.  It  presents  four  years  of  work,  considering  Number, 
Color,   Direction,   Dimension,  Botany,  Minerals,  Form,  Lan- 
guage,   Writing,  Pictures,    Modelling,    Drawing,    Singing, 
Geography,  Zoology,  etc.,  etc. 

10.  There  are  686  pages;  a  large  book  devoted  to  the  realities 
of  school  life,  in  realistic  descriptive  language.    It  is  plain, 
real,  not  abstruse  and  uninteresting. 

11.  It  gives  an  insight  into  real  education,  the  education 
urged  by  Pestalozzi,  Froebel*  Mann«  .Page,  Parker,  etc. 


SEND  ALL  ORDERS  TO 

&  L.  KELLOGG  &  CO.,  NEW  TOEK  &  CHICAGO.    39 

Shaw  and  *DonneWs  School  Devices. 

"  SCHOOL  DEVICES."  A  book  of  ways  and  suggestions  for  teachers. 
By  EDWARD  R.  SHAW  and  WEBB  DONNELL,  of  the  High  School  at 
Yonkers,  N.  Y.  Illustrated.  Dark-blue  cloth  binding,  gold, 
16mo,  289  pp.  Price,  $1.25 ;  to  teachers,  $1.00 ;  by  mail,  9  cents 
extra. 

This  valuable  book  has  just  been  greatly  im- 
proved by  the  addition  of  nearly  75  pages  of 
entirely  new  material. 

&^-A  BOOK  OF  "WAYS"  FOR  TEACHERS.^ 

Teaching  is  an  art;  there  are  "ways  to  do  it."  This  book  is  made 
to  point  out  "  ways,"  and  to  help  by  suggestions. 

1.  It  gives  "ways"  for  teaching  Language,  Grammar,  Reading, 
Spelling,  Geography,  etc.    These  are  in  many  cases  novel;  they  are 
designed  to  help  attract  the  attention  of  the  pupil. 

2.  The  "  ways"  given  are  not  the  questionable  "  ways"  so  often  seen 
practised  in  school-rooms,  but  are  in  accord  with  the  spirit  of  modern 
educational  ideas. 

3.  This  book  will  afford  practical  assistance  to  teachers  who  wish  to 
keep  their  work  from  degenerating  into  mere  routine.    It  gives  them, 
in  convenient  form  for  constant  use  at  the  desk,  a  multitude  of  new 
ways  in  which  to  present  old  truths.    The  great  enemy  of  the  teacher 
is  want  of  interest.    Their  methods  do  not  attract  attention.    There  is 
no  teaching  unless  the're  is  attention.    The  teacher  is  too  apt  to  think 
there  is  but  one  "  way"  of  teaching  spelling ;  he  thus  falls  into  a  rut. 
Now  there  are  many  "ways"  of  teaching  spelling,  and  some  "ways" 
are  better  than  others.    Variety  must  exist  in  the  school-room;  the 
authors  of  this  volume  deserve  the  thanks  of  the  teachers  for  pointing 
out  methods  of  obtaining  variety  without  sacrificing  the  great  end 
sought>-^scholarship.    New  "ways"  induce  greater  effort,  and  renewal 
of  activity. 

4.  The  book  gives  the  result  of  la,rge  actual  experience  in  the  school- 
room, and  will  meet  the  needs  of  thousands  of  teachers,  by  placing  at 
their  command  that  for  which  visits  to  other  schools  are  made,  insti- 
tutes and  associations  attended,  viz.,  new  ideas  and  fresh  and  forceful 
ways  of  teaching.    The  devices  given  under  Drawing  and  Physiology 
are  of  an  eminently  practical  nature,  and  cannot  fail  to  invest  these 
subjects  with  new  interest.    The  attempt  has  been  made  to  present 
only  devices  of  a  practical  character. 

5.  The  book  suggests  "ways"  to  make  teaching  effective;  it  is  not 
simply  a  book  of  new  "ways,"  but  of  "ways"  that  will  produce  good 
results. 


SEND  ALL  ORDERS  TO 

52   E.  L.  KELLOGG  &  CO.,  NEW  YORK  &  CHICAGO. 

IVoodbull's  Simple  Experiments  for  the 

SCHOOL-ROOM.  By  Prof.  JOHN  F.  WOODHULL,  Prof,  of 
«  Natural  Science  in  the  College  for  the  Training  of  Teachers, 
New  York  City,  author  of  "  Manual  of  Home-Made  Appa- 
ratus." Cloth,  16ino.  Price,  50  cents;  to  teachers,  40  cents; 
by  mail,  5  cents  extra. 

This  book  contains  a  series  of  simple,  easily-made  experiments, 
to  perform  which  will  aid  the  comprehension  of  every-day  phe- 
nomena. They  are  really  the  very  lessons  given  by  the  author  in 
the  Primary  and  Grammar  Departments  of  the  Model  School  in 
the  College  for  the  Training  of  Teachers,  New  York  City. 

The  apparatus  needed  for  the  experiments  consists,  for  the  most 
part,  of  such  things  as  every  teacher  will  find  at  hand  in  a  school- 
room or  kitchen.  The  experiments  are  so  connected  in  logical 
order  as  to  form  a  continuous  exhibition  of  the  phenomena  of 
combustion.  This  book  is  not  a  science  catechism.  Its  aim  is  to 
train  the  child's  mind  in  habits  of  reasoning  by  experimental 
methods. 

These  experiments  should  be  made  in  every  school  of  our 
country,  and  thus  bring  in  a  scientific  method  of  dealing  with 
nature.  The  present  method  of  cramming  children's  minds  with 
isolated  facts  of  which  they  can  have  no  adequate  comprehension 
is  a  ruinous  and  unprofitable  one.  This  book  points  out  the 
method  employed  by  the  best  teachers  in  tlie  best  schools. 

WHAT    IT   CONTAINS. 


I.  Experiments  with  Paper. 

H.          "  Wood. 

HI.  "          a  Candle. 

IV.  "  "          Kerosene. 

V.  Kindling  Temperature. 


VI.  Air  as  an  Agent  in  Combustion. 
VII.  Products  of  Complete     " 
VIII.  Currents  of  Air,  etc.— Ventila- 
IX.  Oxygen  of  the  Air.  [tion. 

X.  Chemical  Changes. 


In  all  there  are  91  experiments  described,  illustrated  by  35 
engravings. 

Jas.  H.  Canfield.  Univ.  of  Kans.,  Lawrence,  says:—"  I  desire  to  say  most 
emphatically  that  the  method  pursued  is  the  only  true  one  in  all  school 
work.  Its  spirit  is  admirable.  We  need  and  must  have  far  more  of  this 
instruction." 

J.  C.  Packard,  Univ.  of  Iowa,  Iowa  City,  says:—"  For  many  years  shut  up 
to  the  simplest  forms  of  illustrative  apparatus,  I  learned  that  the  necessity 
was  a  blessing,  since  so  much  could  be  accomplished  by  home-made  ap- 
paratus—inexpensive and  effective." 

Henry  R.  Russell,  Woodbury,  N.  J.,  Supt.  of  the  Friends  School:— "Ad- 
mirable little  book.  It  is  just  the  kind  of  book  we  need." 

S.  T.  Button,  Supt.  Schools,  New  Haven,  Ct.— "  Contains  just  the  kind  of 
help  teachers  need  in  adapting  natural  science  to  common  schools." 


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